Some Distance Measures of Single Valued Neutrosophic Hesitant Fuzzy Sets and Their Applications to Multiple Attribute Decision Making

Transcription

1 ew Treds eutrosophc Theory ad pplcatos PR ISWS, SURPTI PRMIK *, IHS C. GIRI 3 epartmet of Mathematcs, Jadavpur Uversty, Kolkata, 70003, Ida. E-mal: paldam00@gmal.com *epartmet of Mathematcs, adalal Ghosh.T. College, Papur, P.O.-arayapr, strct-orth 4 Pargaas, West egal, PI-7436, Ida. Correspodg author s E-mal: sura_pat@yahoo.co. 3epartmet of Mathematcs, Jadavpur Uversty, Kolkata,70003, Ida. E-mal: bcgr.umath@gmal.com Some stace Measures of Sgle Valued eutrosophc Hestat Fuzzy Sets ad Ther pplcatos to Multple ttrbute ecso Makg bstract Sgle-valued eutrosophc hestat fuzzy set s a merged form of sgle-valued eutrosophc sets ad hestat fuzzy sets. Ths set s a useful tool to hadle mprecse, complete ad cosstet formato exstg mult-attrbute decso makg problems. I mult-attrbute decso makg, dstace measures play a mportat role to take a decso regardg alteratves. I ths paper we propose a varety of dstace measures for sgle valued eutrosophc sets. Furthermore, we apply these measures to mult-attrbute decso makg problem wth sglevalued eutrosophc hestat fuzzy set evromet to fd out the best alteratve. We provde a llustratve example to valdate ad to show frutfuless of the proposed approach. Fally, we compare the proposed approach wth other exstg methods for solvg mult-attrbute decso makg problems. Keywords Hestat fuzzy sets, sgle-valued eutrosophc set, sgle-valued eutrosophc hestat fuzzy set, dstace measure, mult-attrbute decso makg problem.. Itroducto stace ad smlarty measures are sgfcat a varety of scetfc felds such as decso makg, patter recogto, ad market predcto. Lots of studes have bee doe o fuzzy sets [], tutostc fuzzy sets [], ad eutrosophc sets [3]. mog them the most wdely used dstace measure are Hammg dstace ad Eucldea dstace. Geerally whe people make decso, they ofte hestate to select for oe thg or aother to reach the fal decso. Tora ad arukawa [4], Tora [5] troduced hestat fuzzy set (HFS), a exteso of fuzzy set, whch allows the membershp degree to assume a set of possble values. HFS ca express the hestat formato compressvely tha other extesos of fuzzy sets. Xu ad Xa [6] defed some dstace measures o the bass of well-kow Hammg dstace ad Eucldea dstace ad the Housdroff metrc. They developed a class of hestat dstace measures ad dscussed some of ther propertes. Peg et al. [7] proposed the geeralsed hestat fuzzy syergetc weghted dstace measure ad appled 7

2 Floret Smaradache, Surapat Pramak (Edtors) t to mult-attrbute decso makg (MM) problem, where the best alteratve. Havg defed hestacy degree, L et al. [8] proposed some dstace ad smlarty measures o HFSs ad developed a TOPSIS method for MM. O the other had, Zhu et al. [9] proposed a dual hestat fuzzy set (HFS) whch cossts of two parts oe s the membershp hestacy fucto ad aother s the o-membershp hestacy fucto. HFS geeralses fuzzy set (FS), tutostc fuzzy set (IFS), hestat fuzzy set (HFS), ad ts membershp degree ad o-membershp degree are preseted by two set of possble values. Cosequetly, HFS ca represet mprecse ad ucerta formato exstg real decso makg problem more flexble way tha FS, IFS, HFS. Sgh [0] defed some dstace ad smlarty measures of HFSs o the bass of the geometrc dstace model, the set theoretc approach ad the matchg fuctos to study MM wth HFSs. However, HFSs ad HFSs caot represet determacy hestat fucto for complete or cosstet formato. Ths type of fucto s a aother ssue to be cosdered decso makg ad thus t should be cluded wth membershp hestat ad o-membershp hestat fucto to catch up mprecse, complete, cosstet formato foud decso makg process. Ye [] troduced sgle-valued eutrosophc hestat fuzzy set (SVHF) whch cossts of three parts the truth membershp hestacy fucto, the determacy membershp hestacy fucto, ad falsty membershp hestacy fucto. Ths set ca express mprecse, complete, cosstet formato wth these three kds of hestacy fuctos a more flexble way. I same dscussos [], Ye developed two aggregato operators for SVHFS formato ad appled these operators to MM problems. Sah ad Lu [] defed correlato coeffcet of SVHFSs to solve MM wth SVHFSs. Lterature revew suggests that the dstace measures ad smlarty measures have ot bee studed, therefore we eed to develop dstace measures for SVHFSs. I ths paper, we propose a class of dstace measures for sgle-valued eutrosophc hestat fuzzy sets ad study ther propertes wth varatoal parameters. We apply the weghted dstace measures to calculate the dstaces betwee each alteratve ad deal alteratve the MM problems. Wth these dstace values, we preset the rakg order of alteratves for selectg the best oe. We preset a llustratve example to verfy the proposed approach ad to show ts frutfuless. Fally, we compare the proposed method wth other exstg methods for solvg MM uder SVHF evromet. The rest of the paper s orgased as follows: Secto presets some bascs of sgle-valued eutrosophc set ad hestat fuzzy sets ad the exstg dstace measures for HFSs. Secto 3 proposes Hammg dstace measure, Eucldea dstace measure, geeralsed dstace measure, ad Hausdroff dstace. Secto 4 devotes applcato of proposed dstace measure to MM wth SVHFS formato. I Secto 5, a llustratve example s gve to valdate ad show effectveess of the proposed approach. I Secto 6, we preset cocludg remarks ad future scope of research.. Prelmares I ths secto we revew some basc deftos regardg sgle-valued eutrosophc sets ad hestat fuzzy sets to develop the preset paper. 8

3 ew Treds eutrosophc Theory ad pplcatos.. Sgle valued eutrosophc set efto. [3] Let X be a space of pots (obects) wth a geerc elemet X deoted by x. SVS X s characterzed by a truth membershp fucto T (x), a determacy membershp fucto I (x), ad a falsty membershp fucto F (x) ad s deoted by Here T (x), I (x) ad (x) x, T (x), I (x), F (x) x X. F are real subsets of [0,] that s (x) : X [0,] T, (x) : X [0,] T I F For I ad F (x) : X [0,]. The sum of T (x), I (x) ad F (x) les [0, 3] that s 0 (x) (x) (x) 3. coveece, SVS ca be deoted by T (x), I (x), F (x) for all x X. ow we meto some commoly used dstace measures for two SVS ad o X x x x {,,..., }.. ormalzed Hammg dstace measure [4]: (, ) T ( x ) T ( x ) I ( x ) I ( x ) F ( x ) F ( x ) Ham 3 (). ormalzed Eucldea dstace measure [4]: (, ) T ( x ) T ( x ) I ( x ) I ( x ) F ( x ) F ( x ) Euc 3 3. The Hausdroff metrc [5]: () (, ) max T ( x ) T ( x ), I ( x ) I ( x ), F ( x ) F ( x ).. Hestat fuzzy sets Ham (3) efto. [4, 5, 6] Let X be a fxed set. hestat fuzzy set o X s preseted terms of a fucto such that whe appled to X returs a subset of[0,],.e. x, h ( x) x X, where h ( x) s a set of some dfferet values [0,], represetg the possble membershp degrees of the elemet x X to. For coveece, h( x ) s called a hestat fuzzy elemet (HFE). We have some well-kow dstace measures for two SVS ad o X { x, x,..., x }.. Geeralzed hestat ormalzed dstace: h h lx ( ) ( ) G (, ) lx /, 0 (4). Geeralzed hestat ormalzed Hausdorff dstace: / ( ) ( ) Hau (5) (, ) max h h, 0. lx max l( h ( )), ( ( )) x l h x for each x X ; () h ( x) ad () h ( x) are the th largest values h( x ) ad h ( x ), respectvely. l( h ( x )) ad l( h ( x )) are the umber of values h ( x ) ad h ( x ), respectvely. efto 3. [6] Let, ad 3 be three HFSs o X {x, x,..., x }, the the dstace measure betwee ad s defed as d, whch satsfes the followg propertes:, 9

4 Floret Smaradache, Surapat Pramak (Edtors). 0 d, ;. d, 0f ad oly f 3. d, = d, ; ; The smlarty measure betwee ad s defed as propertes:. 0 s, ;. s, f ad oly f ; 3. s, = s,. s,, whch satsfes the followg If d, be the dstace measure betwee two HFSs ad, the s, d, s the smlarty measure betwee two HFSs ad. Smlarly, f s, be the smlarty measure betwee two HFSs ad, the d, s, s the dstace measure betwee two HFSs ad. 3. stace measure of sgle valued eutrosophc sets The eutrosophc set [3] theory poeered by Smaradache has emerged as oe of the research focus may braches such as maagemet sceces, egeerg, appled mathematcs. eutrosophc set geeralzes the cocept of the crsp set, fuzzy set [], terval valued fuzzy set [7], tutostc fuzzy set [], ad terval valued tutostc fuzzy set [8]. efto 4. [] Let X be a fxed set, the a sgle valued eutrosophc hestat fuzzy set o X s defed as follows: x, t( x), ( x), f ( x) x X, where, tx ( ), x ( ), f( x ) are three sets of some values [0,], deotg the respectvely the possble truth, determacy ad falsty membershp degrees of the elemet x X to the set. The membershp degrees tx ( ), x ( ) ad f( x ) satsfy the followg codtos: 0,,, 0 3 where, tx ( ), x ( ), f( x), ad x ( ) t( x) f ( x) max f ( x) for all x X. t ( x) max t( x), x ( ) ( x) max ( x) For coveece of otato, the trple ( x) t( x), ( x), f ( x) s called a sgle valued eutrosophc hestat fuzzy elemet (SVHFE) ad s deoted by t,, f. It s to be oted that the umber of values for possble truth, determacy ad falsty membershp degrees of the elemet dfferet SVHFEs may be dfferet. efto 5. [] Let t,, f ad t,, f be two SVHFEs, the followg operatoal rules are defed as follows:. t t t t f f { },{ },{, } ; t,, f, t,, f 30

5 ew Treds eutrosophc Theory ad pplcatos. t t f f f f { },{ },{ } ; t,, f, t,, f 3. t f { ( ) },{ },{ }, 0 ; t,, f 4. t f { },{ ( ) },{ ( ) }, 0. t,, f Motvatg from the cocept provded by Xu ad Xa [6], we defe a geeralzed sgle valued eutrosophc hestat ormalzed dstace: # hx # gx # m x ( ) ( ) ( k ) ( k ) ( p) ( p) G ( x ) ( x ) ( x ) ( x ) ( x ) ( x ), 0 3 l x k p (6) where # # # If, lx h x g x m x ; # h x, # g ad # m are the umber of elemets t,, ad f, respectvely. x x Eq.(9) reduces to sgle valued eutrosophc hestat ormalzed Hammg dstace: # hx # gx # m x ( ) ( ) ( k) ( k) ( p) ( p) GHam ( x ) ( x ) ( x ) ( x ) ( x ) ( x ) 3 l x k p If, (7) Eq.(9) reduces to sgle valued eutrosophc hestat ormalzed Eucldea dstace: x x x x x x # hx # gx # m x ( ) ( ) ( k ) ( k) ( p) ( p) G ( ) ( ) ( ) ( ) ( ) ( ) 3 l x k p (8) However, f we cosder, Hausdroff metrc to the dstace measure, the the geeralzed sgle valued eutrosophc hestat ormalzed Hausdorff dstace ca be defed as follows: GHau ( ) ( ) ( k ) ( k ) max ( x ) ( x ),max ( x ) ( x ), k (, ) max, 0. ( p) ( p) max ( x ) ( x ) p (9) For, Eq.() reduces to sgle valued eutrosophc hestat ormalzed Hammg Hausdorff dstace: GHau ( ) ( ) ( k ) ( k ) max ( x ) ( x ),max ( x ) ( x ), k (, ) max. 3 ( p) ( p) max ( x ) ( x ) p (0) For, Eq.() reduces to sgle valued eutrosophc hestat ormalzed Eucldea Hausdorff dstace: GHau ( ) ( ) ( ) ( ) k k max ( x ) ( x ),max ( x ) ( x ), k (, ) max. ( ) ( ) 3 p p max ( x ) ( x ) p () I some stuatos, we eed weght of each elemet x X, ad the we preset the followg weghted dstace measures for SVHFs. ssume that the weght of the elemet x X s w (,,..., ) wth w [0,] ad w, the we have a geeralzed SVHF weghted dstace: # hx # gx # m x w ( ) ( ) ( k) ( k) ( p) ( p) G ( x ) ( x ) ( x ) ( x ) ( x ) ( x ), 0 3 l x k p () ad a geeralzed SVH weghed Hausdroff dstace: / / / / / 3

6 Floret Smaradache, Surapat Pramak (Edtors) GHau ( ) ( ) ( k ) ( k ) max w ( x ) ( x ),max w ( x ) ( x ), k (, ) max, 0. 3 ( p) ( p) max w ( x ) ( x ) p (3) 4. pplcato of proposed dstace measure mult-attrbute decso makg I ths secto we use the proposed dstace measures to fd out the best alteratve multattrbute decso makg wth sgle valued eutrosophc hestat fuzzy evromet. For a mult-attrbute decso makg problem, assume that,,..., m be the set of m alteratves, C { C, C,..., C } be the set of attrbutes, whose weght vector w ( w, w,..., w ) T, satsfes w 0(,,..., ) ad w, where w deotes the weght of the attrbute C. The performace of the alteratve wth respect to the attrbute C s measured by a SVHFE { t,, f}, where t { t,0 }, {,0 }, ad f { f,0 } are the possble truth, determacy ad falsty membershp degree, respectvely such that 0 3, where, max{ }, max{ }.) t, ad max{ } ll { t,, f }(,,..., m;,,..., ). f / are cotaed SVHF decso matrx ( ) (See Table Table. SVHF decso matrx C C C m m m m m ascally attrbutes are two types:. beeft type attrbutes,. cost type attrbutes. * * * * * I such cases, we propose the ratg values of deal alteratves as { t,, f } where, * * {{},{0},{0}} for beeft type attrbutes ad {{0},{},{}} for cost type attrbutes. The to determe the best alteratves, we propose the followg steps: Step. eterme the dstace betwee a alteratve (,,..., ) ad the deal alteratve usg proposed dstace measure accordg to the ature of attrbutes. Step. Rak the alteratve o the bass of dstace measure values. Step 3. Obta the best alteratve accordg to the mmum value of dstace measure. for,,...,, 5. umercal example I ths secto we cosder the example adopted from Ye [] to llustrate the applcato of the proposed GR method for MM proposed Secto 4. Cosder a vestmet compay that wats to vest a sum of moey the best opto. There s a pael wth four possble alteratves: () s the car compay; () s the food compay; (3) 3 s the computer compay; (4) 4 s * 3

7 ew Treds eutrosophc Theory ad pplcatos the arms compay. To take a decso, the vestmet compay cosder three attrbutes: () C s the rsk aalyss; () C s the growth aalyss; (3) C 3 s the evrometal mpact aalyss. The attrbute weght vector s gve as W (0.35, ) T. The four possble alteratves {,, 3, 4} are evaluated by usg SVHFEs uder three attrbutes C(,,3). We ca arrage the ratg values a matrx form amely a SVHF decso matrx X ( x) that s show Table- 43. Table. Sgle valued eutrosophc hestat fuzzy decso 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 ow we cosder the followg steps, descrbed Secto-4, to fd the best alteratves. Step.Usg Eq.(4), we calculate the SVH weghted dstace measure betwee alteratves * (,,3,4) ad deal alteratve for,,5,0 whch are show Table.: Table. Weghted dstat measures for dfferet values of s Rakg = = =5 = Step.Rak the alteratve accordg to the value of SVH weghted dstace measure. Step 3. ased o the mmum value of SVH weghted dstace measures for dfferet values of,,5,0, we coclude that as the best alteratve, whch s same as the results obtaed Ye [] ad Sah ad Lu []. From Table-, we observe that rakg results chage wth dfferet values of.therefore takg the values of accordg to decso maker s preferece play a crucal role rakg process. Ye [] cosdered weghted cose smlarty measure of SVHFs ad Sah ad Lu [] proposed weghted correlato co-effcet to determe the rakg order of alteratves. I both studes, we see that there s o opto to cosder dfferet values of the atttudal character that ca chage the rakg result as we have see our study. Thus our method s more realstc ad flexble over these two methods, furthermore our method s smple ad effectve. 6. Cocluso I ths paper, we develop a class of dstace measures for sgle-valued eutrosophc hestat fuzzy sets ad dscussed ther propertes wth varatoal parameters. We apply the weghted 33

8 Floret Smaradache, Surapat Pramak (Edtors) dstace measures to calculate the dstaces betwee each alteratve ad deal alteratve the MM problems. Wth these dstace values, we obta the rakg order of alteratves for selectg the best oe. We provde a llustratve example to verfy the proposed approach ad to show ts frutfuless. Fally, we compared the proposed method wth other exstg methods for solvg MM uder SVHF evromet. The proposed method s smple ad effectve to hadle MM uder SVHF. We hope that the proposed dstace measures ca be exteded to terval eutrosophc hestat fuzzy set, ad ca be appled medcal dagoss, patter recogto, ad persoal selecto uder eutrosophc hestat fuzzy evromet. Refereces. L.. Zadeh, Fuzzy sets, Iformato Cotrol, 8(965) K.T. taassov, Itutostc fuzzy sets, Fuzzy Sets ad Systems 0(986) F. Smaradache, ufyg feld logcs, eutrosophy: eutrosophc probablty, set ad logc. merca Research Press, Rehoboth, V. Torra, Y. arukawa, O hestat fuzzy sets ad decso : The 8th IEEE Iteratoal Coferece o Fuzzy Systems, Jeu Islad, Korea, V. Torra, Hestat fuzzy sets, Iteratoal Joural of Itellget Systems 5(00) M.M. Xa, Z.S. Xu, Hestat fuzzy formato aggregato decso makg, Iteratoal Joural of pproxmate Reasog 5(0) H. Peg, C.Y. Gao, Z. F. Gao, Geeralzed hestat fuzzy syergetc weghted dstace measures ad ther applcato to multple crtera decso-makg, ppled Mathematcal Modellg 37(8)(03) L. Wag, S. Xu, Q. Wag, M.. stace ad smlarty measures of dual hestat fuzzy sets wth ther applcatos to multple attrbute decso makg. I Progress Iformatcs ad Computg (PIC), 04 Iteratoal Coferece o 04 May 6 (88-9). IEEE. 9.. Zhu, Z.S. Xu, M.M. Xa, ual hestat fuzzy sets, Joural of ppled Mathematcs (0) do: 0.55/0/ P. Sgh, stace ad smlarty measures for multple-attrbute decso makg wth dual hestat fuzzy sets. Computatoal ad ppled Mathematcs (03) do: 0.007/s J. Ye, Multple-attrbute decso makg uder a sgle-valued eutrosophc hestat fuzzy evromet, Joural of Itellget Systems (04) do: 0.55/sys R. Sah, P Lu, Correlato coeffcet of sgle-valued eutrosophc hestat fuzzy sets ad ts applcatos decso makg, eural Computg ad pplcatos (06) do: 0.007/s x. 3. H.Wag, F. Smaradache, R. Suderrama, Y.Q. Zhag, Sgle-valued eutrosophc sets, Mult space ad Mult structure. 4(00) P. Maumdar, S.K. Samata, O smlarty ad etropy of eutrosophc sets, Joural of Itellget ad fuzzy Systems, 6(3)(04) S. roum,f. Smaradache, Several smlarty measures of eutrosophc sets. eutrosophc Sets ad Systems, ()(03), Z. Xu,M. Xa, stace ad smlarty measures for hestat fuzzy sets. Iformato Sceces, 8()(0), M.. Gorzalczay, method of ferece approxmate reasog based o terval-valued fuzzy sets, Fuzzy Sets ad Systems (987) K. taassov, G. Gargov, Iterval valued tutostc fuzzy sets. Fuzzy sets ad systems, 3(3) (989),