Multi criteria decision making using correlation coefficient under rough neutrosophic environment

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1 Neutrosophc Sets ad Systems Vol Uversty of New Mexco Mult crtera decso makg usg correlato coeffcet uder rough eutrosophc evromet Surapat Pramak Rum Roy Tapa Kumar Roy 3 ad Floret Smaradache 4 Departmet of Mathematcs Nadalal Ghosh.T. College Papur PO-Narayapur ad Dstrct: North 4 Pargaas P Code: 7436 West egal Ida. Emal: sura_pat@yahoo.co. Departmet of Mathematcs Ida Isttute of Egeerg Scece ad Techology Shbpur Howrah-703 West egal Ida. Emal: roy.rum.r@gmal.com 3 Departmet of Mathematcs Ida Isttute of Egeerg Scece ad Techology Shbpur Howrah-703 West egal Ida. Emal: roy_t_k@gmal.com 4 Uversty of New Mexco. Mathematcs & Scece Departmet 705 Gurley ve. Gallup NM 8730 US. Emal: fsmaradache@gmal.com bstract I ths paper we defe correlato coeffcet measure betwee ay two rough eutrosophc sets. We also prove some of ts basc propertes.. We develop a ew multple attrbute group decso makg method based o the proposed correlato coeffcet measure. llustratve example of medcal dagoss s solved to demostrate the applcablty ad effecrveess of the proposed method. Keywords: Mult-attrbute group decso makg; Neutrosophc set; Rough set; Rough eutrosophc set; Correlato coeffcet. Itroducto Smaradache establshed the cocept of eutrosophc set ad eutrosophc logc [] to deal ucertaty cosstecy completeess ad determacy 998. Smaradache [] ad Wag et. al. [] studed sgle valued eutrosophc set (SVNS a subclass of eutrosophc set to deal realstc problems 00. SVNSs have bee wdely studed ad appled dfferet felds such as medcal dagoss [3] mult crtera decso makg [ ] mage processg [8 9 0] etc. Pawlak [] defed rough set to study tellgece systems characterzed by exact ucerta or suffcet formato. roum et al. [ 3] defed rough eutrosophc set by combg the rough set ad sgle valued eutrosophc set to deal wth problems volvg ucerta mprecse complete ad cosstet formato exstg real world problems. Decso makg rough eutrosophc evromet s a ew subfeld of operatoal resesarch. I rough eutrosophc evromet Modal ad Pramak [4] defed accumulated geometrc operator to trasform rough eutrosophc umber (eutrosophc par) to sgle valued eutrosophc umber ad developed a ew multattrbute decso-makg (MDM) method based o grey relatoal aalyss. Modal ad Pramak [5] defed accuracy score fucto ad proved ts basc propertes. I the same study Modal ad Pramak [5] preseted a ew MDM method rough eutrosophc evromet. Pramak ad Modal [6] defed cotaget smlarty measure of rough eutrosophc sets ad proved ts basc propertes. I the same study Pramak ad Modal [6] preseted ts applcato to medcal dagoss. Pramak ad Modal [7] proposed cose smlarty measure of rough eutrosophc sets ad ts applcato medcal dagoss. Pramak ad Modal [8] also proposed Dce ad Jaccard smlarty measures rough eutrosophc evromet ad appled them for MDM. Modal ad Pramak [9] studed cose Dce ad Jaccard smlarty measures for terval rough eutrosophc sets ad preseted MDM methods based o proposed rough cose Dce ad Jaccard smlarty measures terval rough eutrosophc evromet Modal et al. [30] preseted rough trgoometrc Hammg smlarty measures such as cose se ad cotaget rough smlarty measures ad proved ther basc propertes. I the same study Modal et al. [30] preseted ew MDM methods based o cose se ad cotaget rough smlarty measures wth llustratve example. Modal et al. [3] proposed varatoal coeffcet smlarty measures uder rough eutrosophc evromet ad proved some of ther basc propertes. I the same study Modal et al. [3] developed a ew MDM method based o the proposed varatoal coeffcet smlarty measures ad preseted a comparso wth four exstg rough smlarty measures amely rough cose smlarty measure rough dce Surapat Pramak Rum Roy Tapa Kumar Roy Floret Smaradache. Mult crtera decso makg usg correlato coeffcet uder rough eutrosophc evromet

2 30 Neutrosophc Sets ad Systems Vol smlarty measure rough cotaget smlarty measure ad rough Jaccard smlarty measure for dfferet values of the parameter. Modal et al. [3] proposed rough eutrosophc aggregate operator ad weghted rough eutrosophc aggregate operator to develop TOPSIS based MDM method rough eutrosophc evromet. Pramak et al. [33] defed projecto ad bdrectoal projecto measures betwee rough eutrosophc sets. I the same study Pramak et al. [33] proposed two ew mult crtera decso makg (MCDM) methods based o eutrosophc projecto ad bdrectoal projecto measures respectvely. Modal ad Pramak [34] proposed rough tr-complex smlarty measure based MDM method rough eutrosophc evromet ad proved some of ts basc propertes. I the same study Modal ad Pramak [34] preseted comparso of obtaed results for a llustratve MDM problem wth other exstg rough eutrosophc smlarty measures. Modal et al. [35] defed rough eutrosophc hypercomplex set ad rough eutrosophc hyper-complex cose fucto ad proved some of ther basc propertes. I the same study Modal et al. [35] also proposed rough eutrosophc hyper-complex smlarty measure based MDM method. Pramak ad Modal [36] defed bpolar rough eutrosophc sets ad proved t basc propertes. The correlato coeffcet s a mportat tool to judge the relato betwee two objects. The correlato coeffcets [ ] have bee wdely employed to data aalyss ad classfcato decso makg patter recogto ad so o. May researchers pay atteto to correlato coeffcets uder fuzzy evromets. Chag ad L [43] troduced the correlato of fuzzy sets. Hog [44] proposed fuzzy measures for a correlato coeffcet of fuzzy umbers uder Tw (the weakest t-orm)-based fuzzy arthmetc operatos. s a exteso of fuzzy correlatos Wag ad L [45] troduced the correlato ad formato eergy of terval-valued fuzzy umbers. Gerstekor ad Mako [46] developed the correlato coeffcets of tutostc fuzzy sets IFSs). Hug ad Wu [47] also proposed a method to calculate the correlato coeffcets of IFSs by cetrod method. Xu [48] developed aother correlato measure of terval-valued tutostc fuzzy evromet ad appled t to medcal dagoss. Ye [49] studed the fuzzy decso-makg method based o the weghted correlato coeffcet uder tutostc fuzzy evromet. ustce ad urllo [50] ad Hog [5] further developed the correlato coeffcets for terval-valued tutostc fuzzy sets (IVIFSs). Haafy et al. [5] troduced the correlato of eutrosophc data. Ye [53] preseted the correlato coeffcet of SVNSs based o the exteso of the correlato coeffcet of IFSs ad proved that the cose smlarty measure of SVNSs s a specal case of the correlato coeffcet of SVNSs. Haafy et al. [54] preseted the cetrod-based correlato coeffcet of eutrosophc sets ad vestgated ts propertes. roum ad Smaradache [55] defed correlato coeffcet of terval eutrosophc set ad vestgated ts propertes. I the lterature o studes have bee reported o MDM usg correlato coeffcet uder rough eutrosophc evromet. To fll the research gap we propose correlato coeffcet uder rough eutrosophc evromet ad proved some of ts basc propertes. We also preset a ew MDM method based o proposed measure. We also preset a llustratve umercal example to show the effectveess ad applcablty of the proposed method. Rest of the paper s orgazed as follows: Secto descrbes prelmares of eutrosophc sets SVNSs ad rough eutrosophc set (RNS). Secto 3 descrbes the correlato coeffcet betwee SVNSs. Secto 4 presets defto ad propertes of proposed correlato coeffcet betwee RNSs. Secto 5 presets a rough eutrosophc decso makg method based o correlato coeffcet. Secto 6 presets a llustratve hypothetcal medcal dagostc problem based o the proposed MDM method. Fally secto 7 presets cocludg remarks ad future scope of research. Prelmares. Neutrosophc sets I 998 Smaradache offered the followg defto of eutrosophc set(ns)[]. Defto.. [] Let X be a space of pots(objects) wth geerc elemet X deoted by x. NS X s characterzed by a truthmembershp fucto T a determacy membershp fucto I ad a falsty membershp fucto F. The fuctos T I ad F are real stadard or o-stadard subsets of ]0 - + [ that s T : X ]0 [ I : X ]0 [ ad. It should be oted that there s o restrcto o the sum of T I ad F.e 0 T I F 3. Defto.. [] (Complemet) The complemet of a eutrosophc set s deoted by C() ad s defed by T c() )={ + }-T I c() )={ + }-I F c() )={ + }-F ). Defto..3 [] eutrosophc set s cotaed aother eutrosophc set deoted by ff f T ) f T sup T ) sup T f I ) f I sup I ) f I f F ) f F ) ad sup F ) sup F ) for all x X. Defto..4 [] Let X be a uversal space of pots (objects) wth a geerc elemet of X deoted by x. sgle valued eutrosophc set s characterzed by a truth membershp fucto T a falsty membershp fucto F ) ad Surapat Pramak Rum Roy Tapa Kumar Roy Floret Smaradache. Mult crtera decso makg usg correlato coeffcet uder rough eutrosophc evromet

3 Neutrosophc Sets ad Systems Vol determacy fucto I ) wth T I ) ad F ) [0] for all x X. Whe X s cotuous a SNVS ca be wrtte as follows: = x T I F ) / x for all x X ad whe X s dscrete a SVNS ca be wrtte as follows : = T I F ) / x for all x X. For a SVNS S0 supt )+ supi )+ supf ) 3. Defto..5 [] The complemet of a sgle valued eutrosophc set s deoted by c() ad s defed by T c() ) = F I c() ) =-I F c() ) = T ). Defto..6 [] SVNS s cotaed the other SVNS deoted as ff T ) T I ) I F ) F ) for all x X.. Rough Neutrosophc sets Rough eutrosophc sets [ 3] are the geeralzato of rough fuzzy sets [ ] ad rough tutostc fuzzy sets [59]. Defto.. [] Let Y be a o-ull set ad R be a equvalece relato o Y. Let P be a eutrosophc set Y wth the membershp fucto T P determacy fucto I P ad omembershp fucto F P. The lower ad the upper approxmatos of P the approxmato space (Y R) are respectvely defed as: xt I / y[x] xy ad R F ) xt I F ) / y[x] R x Y where T ) z [x] R TP (Y I z [x] R IP (Y F ) z [x] F (Y) ad R P T ) z[x] T (YI R P z[x] R IP(YF ) z[x] F (Y) So R P 0 ) ). T ) I ) F ) 3 ad 0 T ) I ) F ) 3. ad deote max ad m operators Here respectvely T P (y I P (y ad F P (y) are the degrees of membershp determacy ad o-membershp of Y wth respect to P. Thus NS mappg N N : N(Y) N(Y) are respectvely referred to as the lower ad upper rough NS approxmato operators ad the par ( N(P ) s called the rough eutrosophc set (Y R). Defto.. [] If = (N(P ) s a rough eutrosophc set (Y R the rough complemet of s the rough eutrosophc set c c c c deoted by ~ () = ((() () ) where (( ) ad () are the complemets of eutrosophc sets N (P) ad respectvely. 3 Correlato coeffcet of SVNSs ased o the correlato of tutostc fuzzy sets Ye [53] defed the formatoal eergy of a SVNS the correlato of two SVNSs ad ad the correlato coeffcet of two SVNSs ad. Defto 3. [53] For a SVNS the uverse of dscourse X = {x x x } the formatoal eergy of the SVNS s defed by I() [T ) I) F )] Defto 3. [53] For two SVNSs ad the uverse of dscourse X = {x x x } correlato of the SVNSs ad s defed as C()= Defto 3. 3 [53] The correlato coeffcet of the SVNSs ad s defed by the followg formula: () C( ) K( )= / [C( ).C( )] [T )T ) I [ [(T )) (I )) (F )) ]] [ = )I ) F )F [(T )) (I )) (F )) ]] The correlato coeffcet K( ) satsfes the followg propertes : () K( ) = K( ); () 0 K( ) ; (3) K( ) = f =. 4 Correlato coeffcet of rough eutrosophc sets Correlato coeffcet betwee rough eutrosophc sets (RNSs) s yet to defe the lterature. Therefore ths paper we defe correlato coeffcet betwee RNSs. Defto4.. ssume that there are ay two RNSs = <( T IF ) ( T IF ) ad = <( T IF ) ( T IF ) ). The the correlato betwee the RNSs ad s defed as C()= [ T ). T ) I ). I ) F ). F where T ) T ) T ) I ) I ) I ) F ) F ) F ) T ) T ) T ) Surapat Pramak Rum Roy Tapa Kumar Roy Floret Smaradache. Mult crtera decso makg usg correlato coeffcet uder rough eutrosophc evromet

4 3 Neutrosophc Sets ad Systems Vol I ) I ) I ) ad F ) F ) F ). Defto 4..The correlato coeffcet of the RNSs ad s defed as C( ) K()= / = [C( ).C( )] [ T ). T ) I ). I ) F ). F...() ( [( T )) ( I )) ( F )) ] )( [( T )) ( I )) ( F )) ] ) The correlato coeffcet K( ) satsfes the followg propertes : () K( ) = K( ); () 0 K() ; (3) K( ) = f =. Proof () C() K() / [C().C()] C() K() / [C().C()] () s C( ) 0 C( ) 0 C( ) 0 so K( ) 0. ccordg to the Cauchy Schwarz equalty: (ab... a b ) (a... a )(b... b ) where a b R for =... (ab... a b ) So (a... a ) (b... b ) Replacg a by T ) ad b by T ) we obta K( ). Therefore 0 K( ). () If = C( ) the K() = K() = / [C( ).C( )] C( ) = C( ) Hece proved. Cosderg = we get the followg: (3) K()= T ). T ) I ). I ) F ). F ) )) ) (( T )) ( I )) ( F (( T )) ( I )) ( F )) ) Whch s the cose smlarty measure betwee two RNSs ad [7]. Weghted correlato coeffcet: Let w = {w w w } be the weght vector of the elemets x ( = ). The the weghted correlato coeffcet betwee ad s defed by the followg formula: = K W ( ) w [ T ). T ) I ). I ) F ). F [( {w [( T )) ( I )) ( F )) ]} ) ( {w [( T )) ( I )) ( F )) ]} )] If w = {/ / /} the equato (4) reduces to equato (). Weghted correlato coeffcet K w ( ) also satsfes the followg propertes: () K w ( ) = K w ( ); () 0 K w () ; (3) K w ( ) = f =. Proof () K W ( ) w [ T ). T ) I ). I ) F ). F [( {w [( T )) ( I )) ( F )) ]} ) ( {w [( T )) ( I )) ( F )) ]} )] w [ T ). T ) I ). I ) F ). F K ( ) w () s [( {w [( T )) ( I )) ( F )) ]} ) ( {w [( T )) ( I )) ( F )) ]} )] {w [( T )) ( I )) ( F )) ]} 0 ad {w [( T )) ( I )) ( F )) ]} 0 (4) w [ T ). T ) I ). I ) F ). F 0 so K W () 0. Usg the weghted Cauchy Schwarz equalty [60] we have (wab... w a b ) (wa... wa )(wb... w b ) where w a b R for =.... (w a b... w a b ) So (w a... w a ) (w b... w b ) Replacg a by w T ) ad b by w T ) we obta K W ( ). Therefore 0 K( ). () If = the K() = K() = w [ T ). T ) I ). I ) F ). F [( {w [( T )) ( I )) ( F )) ]} ) ( {w [( T )) ( I )) ( F )) ]} )] {w [( T )) ( I )) ( F )) ]} {w [( T )) ( I )) ( F )) ]} Hece proved. Surapat Pramak Rum Roy Tapa Kumar Roy Floret Smaradache. Mult crtera decso makg usg correlato coeffcet uder rough eutrosophc evromet

5 Neutrosophc Sets ad Systems Vol Rough eutrosophc decso makg based o correlato coeffcet Let... m be a set of elemets (/objects / persos C C... C be a set of crtera for each elemet ad E E... E k are the alteratves for each elemet. Step. The relato betwee elemets ( =... m) ad the crtera C j (j =... ) s preseted Table terms of RNSs. Table : Relato betwee elemets ad crtera C X X m X m where C X X... X m... C... X... X X m Xj (Tj Ij Fj) (Tj Ij Fj) wth T I F 3 ad 0 T I F 3. 0 j j j j j j The relato betwee crtero C ( =... ) ad the alteratve E j (j =... k) s preseted Table terms of RNSs. Table : Relato betwee crtera ad alteratves E E... Ek C Y Y... Y k C Y Y Y... k C m Y Y... Yk where Y (T I F ) (T I F ) j wth j j j j 0 Tj Ij Fj 3 ad 0 Tj Ij Fj 3. j j Step. Determe the correlato measure betwee Table ad Table usg equato. The obtaed values are preseted Table 3. Table 3 : Correlato coeffcet betwee table ad table E E... Ek p p... p k p p p... k m pm pm... p mk Step 3. From Table 3 for each elemet ( =... m fd the maxmum correlato value of the -th row ( =... m). If the maxmum value occurs at j-th colum ( j =... k) (see Table 3 the E j wll be the best alteratve for the elemet ( =... m). Step 4. Ed. 6 Medcal Dagoss Problem We cosder a medcal dagoss problem for llustrato of the proposed method. Medcal dagoss comprses of cosstet determate ad complete formato though creased volume of formato avalable to doctors from ew medcal techologes. The proposed correlato coeffcets amog the patets versus symptoms ad symptoms versus dseases wll provde medcal dagoss. Let P = {P P P 3 } be a set of patets D = {Vral fever Malara Stomach problem Chest problem} be a set of dseases ad S = {Temperature Headache Stomach pa Cough Chest pa} be a set of symptoms. Usg proposed method the doctor s to exame the patet ad to determe the dsease of the patet rough eutrosophc evromet. ased o the proposed approach the cosdered problem s solved usg the followg steps: Step. Costructo of the rough eutrosophc decso matrx Table 4: (Relato-) The relato betwee Patets ad Symptoms P P P 3 Temperat ure <(.6.4.3) (.8..) <(.5.3.4) (.7.3.) <(.6.4.4) (.8..) Headac he < ( (.6.. ) <( ( ) <( (.7.0. ) Stomac h pa <(.5.3. (.7.. ) <( (.7..4 ) <( (.8.. ) cough <( (.8.0. ) <( (.9..3 ) <( (.8.. ) Chest pa < ( (.6.. ) <( (.7..3 ) <( (.7.. ) Table 5: (Relato-) The relato amog Symptoms ad Dseases Vral Fever Malara Stomach problem Chest problem Temperatu re <( (.8.3.) Headache <( (.7.3.) Stomach pa <(..3.4 (.4.3.) cough <( (.6..) <(..4.4 (.5..) <(..3.4 (.6.3.) <(..4.4 (.3..) <( (.5..3) <( (.5..) <(..3.3 (.4..) <( (.6..) <(..6.6 (.3.4.4) <(..4.6 (.4.4.4) <(..5.5 (.5.3.3) <(..4.6 (.3..4) <( (.7..) Surapat Pramak Rum Roy Tapa Kumar Roy Floret Smaradache. Mult crtera decso makg usg correlato coeffcet uder rough eutrosophc evromet

6 34 Neutrosophc Sets ad Systems Vol Chest pa <(..4.4 (.4..) <(..3.3 (.3..) <(..4.4 (.3..) <( (.6..3) Step. Determato of correlato coeffcet betwee table ad table Table 6: The correlato measure betwee Relato- ad Relato- Vral Fever Malara Stomach problem Chest problem P P P Step 3. Rakg the alteratves ccordg to the values of correlato coeffcet of each alteratve show Table 3 the hghest correlato measure occurs colum(.e. for the dseases vral fever. Therefore all three patets P P P 3 suffer from vral fever. 7 Cocluso I ths paper we have proposed correlato coeffcet ad weghted correlato coeffcet betwee rough eutrosophc sets ad proved some of ther basc propertes. We have developed a ew mult crtera decso makg method based o the correlato coeffcet measure. We preseted a llustratve example medcal dagoss. We hope that the proposed method ca be appled solvg realstc mult crtera group decso makg problems rough eutrosophc evromet. Refereces [] F. Smaradache. ufyg feld logcs: eutrosophc logc. Neutrosophy eutrosophc set eutrosophc probablty ad eutrosophc statstcs Rehoboth: merca Research Press. (998). [] H. Wag F. Smaradache F. Q. Zhag ad R. Suderrama. Sgle valued eutrosophc sets. Multspace ad Multstructure 4 ( [3] J. Ye. Improved cose smlarty measures of smplfed eutrosophc sets for medcal dagoses. rtfcal Itellgece Medce 63 ( [4] J. Ye. Multcrtera decso-makg method usg the correlato coeffcet uder sgle-valued eutrosophc evromet. Iteratoal Joural of Geeral Systems 4 ( [5] J. Ye. Trapezodal eutrosophc set ad ts applcato to multple attrbute decso-makg. Neural Computg ad pplcatos 6 ( [6] P. swas S. Pramak ad. C. Gr. Etropy based grey relatoal aalyss method for mult-attrbute decso makg uder sgle valued eutrosophc assessmets. Neutrosophc Sets ad Systems. ( [7] P. swas S. Pramak ad. C. Gr. Cose smlarty measure based mult-attrbute decso-makg wth trapezodal fuzzy eutrosophc umbers. Neutrosophc Sets ad Systems 8 ( [8] P. swas S. Pramak ad. C. Gr. TOPSIS method for mult-attrbute group decso-makg uder sgle valued eutrosophc evromet. Neural Computg ad pplcatos 7(3) ( do: 0.007/s [9] P. swas S. Pramak ad. C. Gr. ggregato of tragular fuzzy eutrosophc set formato ad ts applcato to mult-attrbute decso makg. Neutrosophc Sets ad Systems. ( [0] P. swas S. Pramak ad. C. Gr. Value ad ambguty dex based rakg method of sgle-valued trapezodal eutrosophc umbers ad ts applcato to mult-attrbute decso makg. Neutrosophc Sets ad Systems ( [] P. swas S. Pramak ad. C. Gr. No-lear programmg approach for sgle-valued eutrosophc TOPSIS method. New Mathematcs ad Natural Computato I Press. []. Kharal. eutrosophc mult-crtera decso makg method. New Mathematcs ad Natural Computg 0 ( do: 0.4/S [3] P. D. Lu ad H. G. L. Multple attrbute decsomakg method based o some ormal eutrosophc oferro mea operators. Neural Computg ad pplcatos 8 ( [4] R. Sah ad P. Lu. Maxmzg devato method for eutrosophc multple attrbute decso makg wth complete weght formato. Neural Computg ad pplcatos (05). do: 0.007/s [5] S. Pramak P. swas ad. C. Gr. Hybrd vector smlarty measures ad ther applcatos to multattrbute decso makg uder eutrosophc evromet. Neural Computg ad pplcatos 8(5) ( [6] K. Modal ad S. Pramak. Neutrosophc decso makg model of school choce. Neutrosophc Sets ad Systems 7 ( [7] K. Modal ad S. Pramak. Neutrosophc taget smlarty measure ad ts applcato to multple attrbute decso makg. Neutrosophc sets ad systems 9 ( [8] Y. Guo ad H. D. Cheg. New eutrosophc approach to mage segmetato. Patter Recogto 4 ( [9] Y. Guo. Segur ad J. Ye. ovel mage thresholdg algorthm based o eutrosophc smlarty score. Measuremet 58 ( Surapat Pramak Rum Roy Tapa Kumar Roy Floret Smaradache. Mult crtera decso makg usg correlato coeffcet uder rough eutrosophc evromet

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