Several Trigonometric Hamming Similarity Measures of Rough Neutrosophic Sets and their Applications in Decision Making

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1 New Tres Neutrosophc Theory a pplcatos KLYN MONDL 1 URPTI PRMNIK 2* FLORENTIN MRNDCHE 3 1 Departmet of Mathematcs Jaavpur Uversty West egal Ia Emal:kalyamathematc@gmalcom ² Departmet of Mathematcs Naalal Ghosh T College Papur PO-Narayapur a Dstrct: North 24 Pargaas P Coe: West egal Ia Correspog author s E-mal: sura_pat@yahooco 3 Mathematcs & cece Departmet Uversty of New Mexco 705 Gurley ve Gallup NM U E-mal: fsmaraache@gmalcom everal Trgoometrc Hammg mlarty Measures of Rough Neutrosophc ets a ther pplcatos Decso Makg bstract I 2014 roum et al ( roum F maraache M Dhar Rough eutrosophc sets Itala Joural of Pure a pple Mathematcs 32 (2014) ) trouce the oto of rough eutrosophc set by combg eutrosophc sets a rough sets whch has bee a mathematcal tool to eal wth problems volvg etermacy a completeess The real worl s full of etermacy Naturally real worl ecso makg problem volves etermacy Rough eutrosophc set s capable of escrbg a halg mprecse etermate a cosstet a complete formato Ths paper s evote to propose several ew smlarty measures base o trgoometrc hammg smlarty operators of rough eutrosophc sets a ther applcatos ecso makg We prove the requre propertes of the propose smlarty measures To llustrate the applcablty of the propose smlarty measures ecso makg a llustratve problem s solve Keywors Neutrosophc set rough set rough eutrosophc set Hammg stace smlarty measure 1 Itroucto L Zaeh [1] trouce the egree of membershp 1965 a efe the cocept of fuzzy set to eal wth ucertaty K T taassov [2] trouce the egree of o-membershp as epeet compoet 1986 a efe the tutostc fuzzy set F maraache [3 4] trouce the egree of etermacy as epeet compoet a efe the eutrosophc set 1998 To use the cocept of eutrosophc set practcal fels such as real scetfc a egeerg applcatos Wag et al [5] presete a stace of eutrosophc set calle sgle value eutrosophc set (VN) 93

2 Floret maraache urapat Pramak (Etors) I may applcatos ue to lack of kowlege or ata about the problem omas the ecso formato may be prove wth tervals stea of real umbers To eal wth the stuato Wag et al [6] trouce terval value eutrosophc sets (IVN) whch s characterze by a membershp fucto o-membershp fucto a a etermacy fucto whose values are tervals rather tha real umbers lso the terval value eutrosophc set ca represet ucerta mprecse complete a cosstet formato whch exst the real worl I 2014 roum et al [7 8] trouce the cocept of rough eutrosophc set (RN) It s erve by hybrzg the cocepts of rough set propose by Pawlak [9] a eutrosophc set orgate by F maraache [3 4] Neutrosophc sets a rough sets are both capable of ealg wth ucertaty a partal formato Rough eutrosophc set [7 8] s the geeralzato of rough fuzzy sets [10] [11] a rough tutostc fuzzy sets [12] Moal a Pramak [13] apple the cocept of rough eutrosophc set mult-attrbute ecso makg base o grey relatoal aalyss 2015 Pramak a K Moal [14] also stue cose smlarty measure of rough eutrosophc sets a ts applcato mecal agoss 2015 Moal a Pramak [15] propose mult attrbute ecso makg usg rough accuracy score fucto Pramak a Moal [16] propose cotaget smlarty measure uer rough eutrosophc evromet Pramak a Moal [17] further propose some smlarty measures amely Dce smlarty measure a Jaccar smlarty measure rough eutrosophc evromet Moal et al [18] propose rough eutrosophc varatoal coeffcet smlarty measure a presete ts applcato mult attrbute ecso makg Moal et al [19] presete rough eutrosophc TOPI for mult-attrbute group ecso makg problem Moal a Pramak [20] stue tr-complex rough eutrosophc smlarty measure a ts applcato mult-attrbute ecso makg Moal et al [21] further propose rough eutrosophc hyper-complex set a ts applcato to mult-attrbute ecso makg Lterature revew reflects that o stues have bee mae o mult-attrbute ecso makg usg trgoometrc Hammg smlarty measures uer rough eutrosophc evromet I ths paper we propose cose se a cotaget Hammg smlarty measures uer rough eutrosophc evromet We also preset a umercal example to show the effectveess a applcablty of the propose smlarty measures 2 Mathematcal Prelmares 21 Neutrosophc set [3 4] Let U be a uverse of scourse The the eutrosophc set s presete the form: = {< x: T( I( F(> x U} where the fuctos T I F: U ] 01 + [ represet respectvely the egree of membershp the egree of etermacy a the egree of omembershp of the elemet xu to the set P satsfyg the followg the coto 0 supt(+ supi(+ supf( gle value eutrosophc sets [6] Defto 22 [6] Wag et al [6] metoe that the eutrosophc set assumes the value from real staar or o-staar subsets of ] [ o stea of ] [ Wag et al [6] coser the terval [0 1] 94

3 New Tres Neutrosophc Theory a pplcatos for techcal applcatos because ] [ s ffcult to apply the real applcatos such as scetfc a egeerg problems ssume that X be a space of pots (obects) wth geerc elemets X eote by x VN X s characterze by a truth-membershp fucto T( a etermacy-membershp fucto I( a a falsty membershp fucto F( for each pot x X T( I( F( [0 1] Whe X s cotuous a VN ca be wrtte as follows: T ( I ( F ( : x X x x Whe X s screte a VN ca be wrtte as follows: T ( I ( F ( 1 : x X x For two VNs VN = {<x: T( I( F(> x X} a VN = {<x T( I( F(> xx } VN VN a VN = VN are efe as follows: (1) VN VN f a oly f T( T( I( I( F(x ) F( (2) VN = VN f a oly f T( = T( I( = I( F( = F( for ay xx 23 Hammg stace [17] Hammg stace [17] betwee two eutrosophc sets T ( I ( F ( a T ( I ( F ( s efe as 1 H ( T ( T ( I ( I ( F ( F ( ) Rough eutrosophc set (RN) Defto 221 [1] [2]: Let Z be a o-ull set a R be a equvalece relato o Z Let be a eutrosophc set Z wth the membershp fucto T etermacy fucto I a omembershp fucto F The lower a the upper approxmatos of the approxmato (Z R) eote by N N N a N are respectvely efe as follows: x T ( I ( F ( / z x x Z N ( N ( N ( x T ( I ( F ( / z x x Z ) where T N ( ( z xr T z I N ( ( z xr I z F N ( z xr F z T ( z xr T z I ( z xr T z F z xr I z R R ( N ( ( o T ( I ( F ( ) 3 a T ( I ( F ( ) 3 hol Here 0 N ( N ( N ( x a eote max a m operators respectvely 0 x ) T z I z a z F (2) are the membershp etermacy a o-membershp egrees of z wth respect to N are two eutrosophc sets Z Thus N mappgs N N : Z) Z) eote respectvely the lower a upper rough N approxmato operators a the par ( ) s calle the rough eutrosophc set (Z R) ase o the above metoe efto t s observe that a N ( have costat membershp o the equvalece class of R f ; e T T ( ) I N ( ( I ( F )( N ( F ( N ( ) N a ( x 95

4 Floret maraache urapat Pramak (Etors) For ay x belogs to Z P s sa to be a efable eutrosophc set the approxmato (Z R) Obvously zero eutrosophc set (0N) a ut eutrosophc sets (1N) are efable eutrosophc sets Defto 222 [1] [2]: Let = ( N ( ) s a rough eutrosophc set (Z R) The c c c c rough complemet of s eote by ~ ( ) where N ( are the complemets of eutrosophc sets of respectvely 96 N N c x F ( 1-I ( T ( / x N ( N ( N ( Z c x F ( 1-I ( T ( / x Z (3) ) Defto 223 [1] [2]: Let N ( a ) Z the the followg eftos hol goo: ) ) ) N ( ) ) ) ) ) ) ) ) ) N ( ) ) ) N ( ) ) ) a are two rough eutrosophc sets respectvely If C are the rough eutrosophc sets (Z R) the the followg propostos ca be state from eftos Proposto 1 [1] [2]: 1 ~ (~ 2 3 ( ) C ( C) ( ) C ( C) 4 ( ) C ( ) ( C) ( ) C ( ) ( C) Proposto 2 [1] [2]: De Morga s Laws are satsfe for rough eutrosophc sets a ) 1 ~ ( )) ( ~ ) (~ )) 2 ~ ( )) (~ ) (~ )) For the proofs of the propostos see [1 2] Proposto 3[1] [2]: If a are two eutrosophc sets U such that 1 ) ) 2 ) ) For the proofs of the propostos see [1 2] Proposto 4 [1] [2]: 1 ~ ~ 2 ~ ~ 3 N ( For the proofs of the propostos see [1 2] 3 Cose Hammg mlarty Measures of RN the N ( ) ssume that T ( I ( F ( T ( I ( F ( T x ) I ( x ) F ( x ) T ( x ) I ( x ) F ( x ) ( a X = {x1 x2 x} be ay two rough

5 New Tres Neutrosophc Theory a pplcatos eutrosophc sets cose Hammg smlarty operator betwee rough eutrosophc sets a s efe as follows: CCHO( )= cos (4) T ( x ) T ( x ) I ( x ) I ( x ) F ( x ) F ( x ) T Here T ( x ) ( T ( T T ( ) 2 x ( T ( 2 I I ( x ) ( I ( I I ( ) 2 x ( I ( 2 F F(x ) ( F ( F F (x ) 2 ( F ( 2 lso [ T ( I ( F ( ] [0 0 0] a [ T ( I ( F ( Proposto 31 ] [0 0 0] = 1 2 The efe rough eutrosophc cose hammg smlarty operator CCHO( ) betwee RNs a satsfes the followg propertes: 1 0 CRCHO ( ) 1 2 CCHO( ) = 1 f a oly f = 3 CCHO( ) = CCHO( Proof of the property 1 ce the fuctos T ( I ( F ( T ( I ( a F ( a the value of the cose fucto are wth [01] the smlarty measure base o rough eutrosophc cose hammg smlarty fucto also les wth [ 01] Hece 0 CCHO ( ) 1 Ths completes thee prove Proof of the property 2 For ay two RNs a f = the the followg relatos hol T ( T ( x ) I ( x ) F x ) F ( x Hece I ) ( ( T ( x ) T ( x ) 0 I ( x ) I ( x ) 0 F ( x ) F ( x ) 0 Thus CCHO( ) = 1 Coversely If CCHO( ) = 1 the T ( x ) T ( x ) 0 I ( x ) I ( x ) 0 F ( x ) F ( x ) 0 sce cos(0) = 1 o we ca wrte x ) T ( x ) Hece = ( T I x ) I ( x ) F x ) F ( x ) ( ( 97

6 Floret maraache urapat Pramak (Etors) 4 e Hammg mlarty Measures of RN ssume that T ( I ( F ( T ( I ( F ( T x ) I ( x ) F ( x ) T ( x ) I ( x ) F ( x ) ( a X = {x1 x2 x} be ay two rough eutrosophc sets se Hammg smlarty operator betwee two rough eutrosophc sets a s efe as follows: CHO( )= 1 1 s T ( T ( I ( I ( F ( F ( (4) 1 6 lso [ T ( I ( F ( ] [0 0 0] a [ T ( I ( F ( ] [0 0 0] = 1 2 Proposto 41 The efe rough eutrosophc se Hammg smlarty operator CHO( ) betwee RNs a satsfes the propertes as follows 1 0 CHO ( ) 1 1 CHO ( ) = 1 f a oly f = 2 CHO( ) = CHO( Proof of the property 1 ce the fuctos T ( I ( F ( T ( I ( a F ( a the value of the se fucto are wth [0 1] the smlarty measure base o rough eutrosophc se hammg smlarty fucto also les wth [ 01] Hece 0 CHO ( ) 1 Proof of the property 2 For ay two RNs a f = the the followg relatos hol T ( T ( x ) I ( x ) F x ) F ( x Hece I ) ( ( T ( x ) T ( x ) 0 I ( x ) I ( x ) 0 F ( x ) F ( x ) 0 Thus CHO ( ) = 1 Coversely If CHO( ) = 1 the T ( x ) T ( x ) 0 I (x ) I (x ) 0 F ( x ) F ( x ) 0 sce s(0) = 0 o we ca wrte x ) T ( x ) Hece = Proof of the property 3 Ths proof s obvous ( T I x ) I ( x ) F x ) F ( x ) ( ( 98

7 New Tres Neutrosophc Theory a pplcatos 5 Cotaget Hammg mlarty Measures of RN ssume that T ( I ( F ( T ( I ( F ( T x ) I ( x ) F ( x ) T ( x ) I ( x ) F ( x ) ( a X = {x1 x2 x} be ay two rough eutrosophc sets cotaget Hammg smlarty operator betwee two rough eutrosophc sets a ca be efe as follows: (5) COTCHO( )= 1 cot T ( x ) T ( x ) I ( x ) I ( x ) F ( x ) F ( x ) lso [ T ( I ( F ( ] [0 0 0] a [ T ( I ( F( ] [0 0 0] = 1 2 Proposto 51 The efe rough eutrosophc cotaget Hammg smlarty operator COTCHO( ) betwee RNs a satsfes the propertes COTCHO ( ) 1 2 COTCHO( ) = 1 f a oly f = 3 COTCHO( ) = COTCHO( Proof of the property 1: Proof: ce the fuctos T ( I ( F ( T ( I ( a F ( a the value of the cotaget fucto are wth [0 1] the smlarty measure base o rough eutrosophc cotagethammg smlarty fucto also les wth [ 01] Hece 0 COTCHO ( ) 1 Proof of the property 2: For ay two RNs a f = we have T x ) T ( x ) I x ) I ( x ) F x ) F ( x ) Hece ( ( ( T (x ) T (x ) 0 I (x ) I (x ) 0 F (x ) F (x ) 0 Coversely If COTCHO( ) = 1 the T (x ) T (x ) 0 ce cot( ) = 1 we ca wrte x ) T ( x ) 4 Hece = Proof of the property 3: Ths proof s obvous Thus COTCHO( ) = 1 I (x ) I (x ) 0 F (x ) F (x ) 0 ( T I x ) I ( x ) F x ) F ( x ) ( ( 99

8 Floret maraache urapat Pramak (Etors) 6 Decso makg uer trgoometrc rough eutrosophc Hammg smlarty measures I ths secto we apply rough cose se a cotaget Hammg smlarty measures betwee RNs to the mult-crtera ecso makg problem ssume that = {1 2 m }be a set of alteratves a ={ 1 2 }be a set of attrbutes The propose ecso makg metho s escrbe usg the followg steps tep 1: Costructo of the ecso matrx wth rough eutrosophc umber Decso maker cosers the ecso matrx wth respect to m alteratves a attrbutes terms of rough eutrosophc umbers as follows 100 Table1: Rough eutrosophc ecso matrx D 1 2 m Here m1 m m m m2 1 2 m 1 2 m s the rough eutrosophc umber accorg to the -th alteratve a the -th attrbute tep 2: Determato of the weghts of attrbute ssume that the weght of the attrbutes ( = 1 2 ) cosere by the ecso-maker be w (( = 1 2 )) such that w [0 1] ( = 1 2 ) a 1 1 w tep 3: Determato of the beeft type attrbute a cost type attrbute Geerally the evaluato attrbute ca be categorze to two types: beeft type attrbute a cost type attrbute Let K be a set of beeft type attrbutes a M be a set of cost type attrbutes I the propose ecso-makg metho a eal alteratve ca be etfe by usg a maxmum operator for the beeft type attrbute a a mmum operator for the cost type attrbute to eterme the best value of each crtero amog all alteratves We efe a eal alteratve * as follows: * = {1* 2* m*} where beeft attrbute s presete as * ( ) ( ) m ( ) max m T I F a cost type attrbute s presete as * ( ) ( ) max ( ) m max T I F tep 4: Determato of the overall weghte rough trgoometrc eutrosophc Hammg smlarty fucto (WRTNHF) of the alteratves We efe weghte rough trgoometrc eutrosophc smlarty fucto as follows CWCHO( ) = 1 w C ( ) (7) CHO WCHO( ) = 1 w ( ) (8) CHO COTWCHO( ) = 1 w COT ( ) (9) CHO (6)

9 New Tres Neutrosophc Theory a pplcatos 1 1 where w = 1 2 tep 5: Rakg the alteratves Usg the weghte rough trgoometrc eutrosophc smlarty measure betwee each alteratve a the eal alteratve the rakg orer of all alteratves ca be eterme a the best alteratve ca be selecte wth the hghest smlarty value tep 6: E 7 Numercal Example ssume that a ecso maker tes to select the most sutable smart phoe for rough use from the three tally chose smart phoes (1 2 3) by coserg four attrbutes amely: features 1 reasoable prce 2 customer care 3 rsk factor 4 ase o the propose approach scusse secto 5 the cosere problem s solve usg the followg steps: tep 1: Costructo of the ecso matrx wth rough eutrosophc umbers The ecso maker forms a ecso matrx wth respect to three alteratves a four attrbutes terms of rough eutrosophc umbers (see the Table 2) Table 2 Decso matrx wth rough eutrosophc umber P) P) (10) tep 2: Determato of the weghts of the attrbutes The weght vectors cosere by the ecso maker are a 012 respectvely tep 3: Determato of the beeft attrbute a cost attrbute Here three beeft types attrbutes a oe cost type attrbute 4 * = [( ) ( ) ( ) ( )] tep 4: Determato of the overall weghte rough trgoometrc eutrosophc Hammg smlarty fucto (WRHNHF) of the alteratves We calculate weghte rough trgoometrc eutrosophc Hammg smlarty values as follows CWCHO(1 * ) = CWCHO(2 * ) = CWCHO(3 * ) = WCHO(1 * ) = WCHO(2 * ) = WCHO(3 * ) = COTWCHO(1 * ) = COTWCHO(2 * ) = COTWCHO(3 * ) = tep 5: Rakg the alteratves Rakg the alteratves s prepare base o the esceg orer of smlarty measures Hghest value reflects the best alteratve Here CWCHO(3 * ) CWCHO(1 * ) CWCHO(2 * ) WCHO(3 * ) WCHO(1 * ) WCHO(2 * ) COTWCHO(3 * ) COTWCHO(1 * ) COTWCHO(2 * ) Hece the smartphoe 3 s the best alteratve for rough use 101

10 Floret maraache urapat Pramak (Etors) tep 6: E 71 Comparso ll the three smlarty measures prove the same rakg orer 8 Cocluso I ths paper we propose rough trgoometrc Hammg smlarty measures base mult-attrbute ecso makg of rough eutrosophc evromet a prove some of ther basc propertes We prove a applcato amely selecto of the most sutable smart phoe for rough use We also preset comparso wth the three rough eutrosophc smlarty measures The cocept presete ths paper ca be apple other multple attrbute ecso makg problems rough eutrosophc evromet Refereces 1 L Zaeh Fuzzy sets Iformato a Cotrol 8(3) (1965) K taassov Itutostc fuzzy sets Fuzzy ets ystems 20(1986) F maraache ufyg fel logcs eutrosophy: eutrosophc probablty set a logc merca Research Press Rehoboth F maraache Neutrosophc set a geeralzato of tutostc fuzzy sets Iteratoal Joural of Pure a pple Mathematcs 24(3) (2005) H Wag F maraache Y Q Zhag R uerrama gle value eutrosophc sets Multspace a Multstructure 4 (2010) H Wag F maraache Y Q Zhag R uerrama Iterval eutrosophc sets a logc: theory a applcatos computg Hexs Phoex Z roum F maraache M Dhar Rough eutrosophc sets Itala Joural of Pure a pple Mathematcs 32 (2014) roum F maraache M Dhar Rough eutrosophc sets Neutrosophc ets a ystems 3 (2014) Z Pawlak Rough sets Iteratoal Joural of Iformato a Computer ceces 11(5) (1982) D Dubos H Prae Rough fuzzy sets a fuzzy rough sets Iteratoal Joural of Geeral ystem 17(1990) Naa Maumar Fuzzy rough sets Fuzzy ets a ystems 45 (1992) K V Thomas L Nar Rough tutostc fuzzy sets a lattce Iteratoal Mathematcs Forum 6(27) (2011) K Moal Pramak Rough eutrosophc mult-attrbute ecso-makg base o grey relatoal aalyss Neutrosophc ets a ystems 7(2015) Pramak K Moal Cose smlarty measure of rough eutrosophc sets a ts applcato mecal agoss Global Joural of vace Research 2(1)(2015) K Moal Pramak Rough eutrosophc mult-attrbute ecso-makg base o rough accuracy score fucto Neutrosophc ets a ystems 8(2015) Pramak K Moal Cotaget smlarty measure of rough eutrosophc sets a ts applcato to mecal agoss Joural of New Theory 4(2015) Pramak K Moal ome rough eutrosophc smlarty measure a ther applcato to mult attrbute ecso makg Global Joural of Egeerg cece a Research Maagemet 2(7)(2015) K Moal Pramak F maraache Mult-attrbute ecso makg base o rough eutrosophc varatoal coeffcet smlarty measure Neutrosophc ets a ystems 13 (2016) (I press) 19 K Moal Pramak F maraache Rough eutrosophc TOPI for mult-attrbute group ecso makg Neutrosophc ets a ystems13 (2016) (I press) 102

11 New Tres Neutrosophc Theory a pplcatos 20 K Moal Pramak (2015) Tr-complex rough eutrosophc smlarty measure a ts applcato mult-attrbute ecso makg Crtcal Revew 11 (2015) K Moal Pramak F maraache Rough eutrosophc hyper-complex set a ts applcato to mult-attrbute ecso makg Crtcal Revew 13 (2016) (I press) 103