Neutrosophic Sets and Systems

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2 SS prt SS X ole eutrosophc Sets ad Systems teratoal Joural formato Scece ad Egeerg Quarterly Edtor--Chef: Prof loret Smaradache ddress: eutrosophc Sets ad Systems teratoal Joural formato Scece ad Egeerg Departmet of Mathematcs ad Scece Uversty of ew Meco 705 Gurley veue Gallup M 8730 US E-mal: smarad@umedu Home page: ssocate Edtor--Chef: Mumtaz l ddress: Uversty of Souther Queeslad 4300 ustrala ssocate Edtors W Vasatha Kadasamy da sttute of echology Chea aml adu da Sad roum Uv of Hassa Mohammeda Casablaca Morocco Salama aculty of Scece Port Sad Uversty Egypt Yahu Guo School of Scece St homas Uversty Mam US racsco Gallego Lupaňez Uversdad Complutese Madrd Spa Pede Lu Shadog Uversty of ace ad Ecoomcs Cha Pabtra Kumar Maj Math Departmet K Uversty W da S lbolw Kg bdulazz Uv Jeddah Saud raba Ju Ye Shaog Uversty Cha Ştefa Vlăduţescu Uversty of Craova Romaa Valer Kroumov Okayama Uversty of Scece Japa Dmtr Rabousk ad Larssa orssova depedet researchers Surapat Pramak adalal Ghosh College West egal da rfa Del Kls 7 ralık Uversty Kls urkey Rıdva Şah aculty of Scece taturk Uversty urkey Luge Vladareau Romaa cademy ucharest Romaa Mohamed bdel-asetaculty of computers ad formatcs Zagazg Uversty Egypt gboola ederal Uversty of grculture beokuta gera Le Hoag So VU Uv of Scece Vetam atoal Uv Hao Vetam Luu Quoc Dat Uv of Ecoomcs ad usess Vetam atoal Uv Hao Vetam Huda E Khald Uversty of elafer elafer - Mosul raq Makel Leyva-Vázquez Uversdad de Guayaqul Guayaqul Ecuador Muhammad kram Uversty of the Pujab Lahore Paksta Volume 3 Cotets 06 K Modal S Pramak ad Smaradache Multattrbute Decso Makg based o Rough eutrosophc Varatoal Coeffcet Smlarty Measure 3 M Malk Hassa S roum Smaradache Regular Sgle Valued eutrosophc Hypergraphs 8 S K De ad eg ragular Dese uzzy eutrosophc Sets 4 H Zaed ad H M agub pplcatos of uzzy ad eutrosophc Logc Solvg Mult-crtera Decso Makg Problems 38 Shah ad S roum rregular eutrosophc Graphs 47 P J M Vera C M Delgado M P Gózalez M L Vázquez Marketg sklls as determats that uderp the compettveess of the rce dustry Yaguach cato pplcato of SV umbers to the prortzato of strateges 70 Smaradache Classcal Logc ad eutrosophc Logc swers to K Georgev 79 M Malk Hassa S roum Smaradache Regular polar Sgle Valued eutrosophc Hypergraphs 84 S Karatas ad C Kuru eutrosophc opology 90 W l-omer eutrosophc crsp Sets va eutrosophc crsp opologcal Spaces C S 96 Salama M Esa H E Ghawalby E awzy eutrosophc eatures for mage Retreval M Sarkar S Dey ad K Roy russ Desg Optmzato usg eutrosophc Optmzato echque 56 6 K Modal S Pramak ad Smaradache Rough eutrosophc OPSS for Mult-ttrbute Group Decso Makg 05 uh era rmal Kumar Mahapatra troducto to eutrosophc Soft Groups 8 Copyrght eutrosophc Sets ad Systems

3 eutrosophc Sets ad Systems Vol 3 06 eutrosophc Sets ad Systems teratoal Joural formato Scece ad Egeerg Copyrght otce eutrosophcs Sets ad Systems ll rghts reserved he authors of the artcles do hereby grat eutrosophc Sets ad Systems o-eclusve worldwde royalty-free lcese to publsh ad dstrbute the artcles accordace wth the udapest Ope tatve: ths meas that electroc copyg dstrbuto ad prtg of both full-sze verso of the joural ad the dvdual papers publshed there for o-commercal academc or dvdual use ca be made by ay user wthout permsso or charge he authors of the artcles publshed eutrosophc Sets ad Systems reta ther rghts to use ths joural as a whole or ay part of t ay other publcatos ad ay way they see ft y part of eutrosophc Sets ad Systems howsoever used other publcatos must clude a approprate ctato of ths joural formato for uthors ad Subscrbers eutrosophc Sets ad Systems has bee created for publcatos o advaced studes eutrosophy eutrosophc set eutrosophc logc eutrosophc probablty eutrosophc statstcs that started 995 ad ther applcatos ay feld such as the eutrosophc structures developed algebra geometry topology etc he submtted papers should be professoal good Eglsh cotag a bref revew of a problem ad obtaed results eutrosophy s a ew brach of phlosophy that studes the org ature ad scope of eutraltes as well as ther teractos wth dfferet deatoal spectra hs theory cosders every oto or dea <> together wth ts opposte or egato <at> ad wth ther spectrum of eutraltes <eut> betwee them e otos or deas supportg ether <> or <at> he <eut> ad <at> deas together are referred to as <o> eutrosophy s a geeralzato of Hegel's dalectcs the last oe s based o <> ad <at> oly ccordg to ths theory every dea <> teds to be eutralzed ad balaced by <at> ad <o> deas - as a state of equlbrum a classcal way <> <eut> <at> are dsjot two by two ut sce may cases the borders betwee otos are vague mprecse Sortes t s possble that <> <eut> <at> ad <o> of course have commo parts two by two or eve all three of them as well eutrosophc Set ad eutrosophc Logc are geeralzatos of the fuzzy set ad respectvely fuzzy logc especally of tutostc fuzzy set ad respectvely tutostc fuzzy logc eutrosophc logc a proposto has a degree of truth a degree of determacy ad a degree of falsty where are stadard or o-stadard subsets of ] [ eutrosophc Probablty s a geeralzato of the classcal probablty ad mprecse probablty eutrosophc Statstcs s a geeralzato of the classcal statstcs What dstgushes the eutrosophcs from other felds s the <eut> whch meas ether <> or <at> <eut> whch of course depeds o <> ca be determacy eutralty te game ukow cotradcto gorace mprecso etc ll submssos should be desged MS Word format usg our template fle: varety of scetfc books may laguages ca be dowloaded freely from the Dgtal Lbrary of Scece: o submt a paper mal the fle to the Edtor--Chef o order prted ssues cotact the Edtor--Chef hs joural s ocommercal academc edto t s prted from prvate doatos formato about the eutrosophcs you get from the UM webste: he home page of the joural s accessed o Copyrght eutrosophc Sets ad Systems

4 eutrosophc Sets ad Systems Vol Uversty of ew Meco Mult-attrbute Decso Makg based o Rough eutrosophc Varatoal Coeffcet Smlarty Measure Kalya Modal Surapat Pramak ad loret Smaradache 3 Departmet of Mathematcs Jadavpur Uversty West egal da Emal: kalyamathematc@gmalcom ²Departmet of Mathematcs adalal Ghosh College Papur PO-arayapur ad Dstrct: orth 4 Pargaas P Code: 7436 West egal da Emal: sura_pat@yahooco 3 Uversty of ew Meco Mathematcs & Scece Departmet 705 Gurley ve Gallup M 8730 US Emal: fsmaradache@gmalcom bstract: he purpose of ths study s to propose ew smlarty measures amely rough varatoal coeffcet smlarty measure uder the rough eutrosophc evromet he weghted rough varatoal coeffcet smlarty measure has bee also defed he weghted rough varatoal coeffcet smlarty measures betwee the rough deal alteratve ad each alteratve are calculated to fd the best alteratve he rakg order of all the alteratves ca be determed by usg the umercal values of smlarty measures ally a llustratve eample has bee provded to show the effectveess ad valdty of the proposed approach Comparsos of decso results of estg rough smlarty measures have bee provded Keywords: eutrosophc set Rough eutrosophc set; Rough varato coeffcet smlarty measure; Decso makg troducto 965 L Zadeh grouded the cocept of degree of membershp ad defed fuzzy set [] to represet/mapulate data wth o-statstcal ucertaty 986 K taassov [] troduced the degree of omembershp as depedet compoet ad proposed tutostc fuzzy set S Smaradache troduced the degree of determacy as depedet compoet ad defed the eutrosophc set [3 4 5] or purpose of solvg practcal problems Wag et al [6] restrcted the cocept of eutrosophc set to sgle valued eutrosophc set SVS sce sgle value s a stace of set value SVS s a subclass of the eutrosophc set SVS cossts of the three depedet compoets amely truthmembershp determacy-membershp ad falstymembershp fuctos he cocept of rough set theory proposed by Z Pawlak [7] s a eteso of the crsp set theory for the study of tellget systems characterzed by eact ucerta or suffcet formato he hybrdzato of rough set theory ad eutrosophc set theory produces the rough eutrosophc set theory [8 9] whch was proposed by roum Dhar ad Smaradache [8 9] Rough eutrosophc set theory s also a powerful mathematcal tool to deal wth completeess Lterature revew reflects that smlarty measures play a mportat role the aalyss ad research of clusterg aalyss decso makg medcal dagoss patter recogto etc Varous smlarty measures [ ] of SVSs ad hybrd SVSs are avalable the lterature he cocept of smlarty measures rough eutrosophc evromet [9 0 ] has bee recetly proposed Pramak ad Modal [9] proposed cotaget smlarty measure of rough eutrosophc sets the same study [9]Pramak ad Modal establshed ts basc propertes ad provded ts applcato to medcal dagoss Pramak ad Modal [0] also proposed cose smlarty measure of rough eutrosophc sets ad ts applcato medcal dagoss he same authors [] also studed Jaccard smlarty measure ad Dce smlarty measures rough eutrosophc evromet ad provded ther applcatos to mult attrbute decso makg Modal ad Pramak [] preseted tr-comple rough eutrosophc smlarty measure ad ts applcato mult-attrbute decso makg ogether wth Smaradache ad S Pramk K Modal [3] preseted hypercomple rough eutrosophc smlarty measure ad ts applcato mult-attrbute decso makg Modal Pramak ad Smaradache [4] preseted several trgoometrc Hammg smlarty measures of rough eutrosophc sets ad ther applcatos mult attrbute decso makg problems Dfferet methods for multattrbute decso makg MDM ad multcrtera decso makg MCDM problems are avalable the lterature dfferet evromet such as crsp evromet [ ] fuzzy evromet [30 3] tutostc fuzzy evromet [ ] eutrosophc evromet [ Kalya Modal Surapat Pramak ad loret Smaradache Mult-attrbute Decso Makg based o Rough eutrosophc Varato Coeffcet Smlarty Measure

5 eutrosophc Sets ad Systems Vol ] terval eutrosophc evromet [ ] eutrosophc soft epert evromet [69] eutrosophc bpolar evromet [70 7] eutrosophc soft evromet [ ] eutrosophc hestat fuzzy evromet [ ] rough eutrolsophc evromet [80 8] etc eutrosophc evromet swas Pramak ad Gr [8] studed hybrd vector smlarty measure ad ts applcato mult-attrbute decso makg Gettg motvato from the work of swas Pramak ad Gr [8] for hybrd vector smlarty measure eutrosophc evomet we eted the cocept rough eutrosophc evromet ths paper a ew smlarty measuremet s proposed amely rough varatoal coeffcet smlarty measure uder rough eutrosophc evromet umercal eample s also provded Rest of the paper s structured as follows Secto presets eutrosophc ad rough eutrosophc prelmares Secto 3 dscusses varous smlarty measures ad varoal coeffcet smlarty measure crsp evromet Secto 4 presets varous smlarty measures ad varatoal smlarty measure for sgle valued eutrosophc sets Secto 5 presets varatoal coeffcet smlarty measure ad weghted varatoal coeffcet smlarty measure for rough eutrosophc sets ad establshes ther basc propertes Secto 6 s devoted to preset mult attrbute decso makg based o rough eutrosophc varatoal coeffcet smlarty measure Secto 7 demostrates the applcato of rough varatoal coeffcet smlarty measures to vestmet problem ally secto 8 cocludes the paper wth statg the future scope of research eutrosophc prelmares Defto [3 4 5] eutrosophc set Let X be a space of pots objects wth geerc elemet X deoted by he a eutrosophc set X : X where s deoted by s the truth membershp fucto s the determacy membershp fucto ad s the falsty membershp fucto he fuctos ad are real stadard or ostadard subsets of ] 0 [ here s o restrcto o the sum of ad e 0 3 Defto [6] Sgle-valued eutrosophc set Let X be a uversal space of pots objects wth a geerc elemet X sgle-valued eutrosophc set SVS X s deoted by / X whe X s cotuous; m / X whe X s dscrete SVS s characterzed by a true membershp fucto a falsty membershp fucto ad a determacy fucto th [0 ] for all X or each X of a SVS 0 3 Some operatoal rules ad propertes of SVSs Let ad be two SVSs X he the followg operatos are defed as follows: Complemet: c - X ddto: Multplcato: - - V Scalar Multplcato: for 0 V for 0 Defto 3 [6] Complemet of a SVS s deoted by c ad s defed by c ; c ; c Defto 4 [6] SVS s cotaed the other SVS deoted as f ad oly f ; ; X Defto 5 [6] wo SVSs ad are equal e = f ad oly f ad Defto 6 [6] Uo of two SVSs ad s a SVS C wrtte as C ts truth membershp determacymembershp ad falsty membershp fuctos are related to those of ad by ma ; m ; C m for all X C Defto 7 [6] tersecto of two SVSs ad s a SVS D wrtte as D whose truth membershp determacy-membershp ad falsty membershp fuctos are related to those of ad by m ; ma C C ; ma for all X C Defto 8 Rough eutrosophc Sets [8 9] Let Z be a o-ull set ad R be a equvalece relato o Z Let P be eutrosophc set Z wth the C Kalya Modal Surapat Pramak ad loret Smaradache Mult-attrbute Decso Makg based o Rough eutrosophc Varato Coeffcet Smlarty Measure

6 eutrosophc Sets ad Systems Vol membershp fucto P determacy fucto P ad o-membershp fucto P he lower ad the upper appromatos of P the appromato Z R deoted by P ad P are respectvely defed as follows: P P P P / z [ ] Z P P P P / z [ ] Z Here P z [ ] R P z P z [ ] R P z z [ ] R P P z [ ] R P z P z P z [ ] R P z P z [ ] R P z So 0 sup P sup P sup P 0 sup P R R 3 sup P sup P 3 Here ad deote ma ad m operators respectvely P z P z ad P z deote respectvely the membershp determacy ad o-membershp fucto of z wth respect to P t s easy to see that P ad P are two eutrosophc sets Z hus S mappgs : Z Z are respectvely referred to as the lower ad the upper rough S appromato operators ad the par P P s called the rough eutrosophc set [8 9] Z R rom the above defto t s see that P ad P have costat membershp o the equvalece classes of R f P = P e P P P P ad P P Z P s sad to be a defable eutrosophc set the appromato Z R t ca be easly proved that zero eutrosophc set 0 = 0 ad ut eutrosophc sets = 0 0 are defable eutrosophc sets Defto 9 [8 9] f P P P s a rough eutrosophc set Z R the rough complemet [8 9] of P s the rough c c eutrosophc set deoted by ~ P P P c where P sets of P P P c P are the complemets of eutrosophc respectvely c P P P P / Z ad c P P P / Z Defto 0 [8 9] f P ad P are the two rough eutrosophc sets of the eutrosophc set P respectvely Z the the followg deftos [8 9] hold: P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P f M L are the rough eutrosophc sets Z R the the followg proposto are stated from deftos [8 9] Proposto [8 9] ~ ~ M M M M 3 L M L M L M L M 4 L M L M L L M L M L Proposto [8 9] De Morga s Laws are satsfed for rough eutrosophc sets ~ P P ~ P ~ P ~ P P ~ P ~ P Proposto 3 [8 9] f P ad P are two eutrosophc sets U such that P P the P P P P P P P P P P Proposto 4 [8 9] P ~ ~ P P ~ ~ P 3 P P 3 Smlarty measures ad varatoal coeffcet smlarty measure crsp evromet he vector smlarty measure s oe of the mportat tools for the degree of smlarty betwee objects However the Jaccard Dce ad cose smlarty measures are ofte used for ths purpose Jaccard [83] Dce [84] ad cose [85] smlarty measures betwee two vectors are stated below Let X = ad Y = y y y be two - dmesoal vectors wth postve co-ordates Defto 3 [83] Jeccard de of two vectors measurg the smlarty of these vectors ca be defed as follows: JX Y = where X Y y = Y -X Y y - y ad Y = y Kalya Modal Surapat Pramak ad loret Smaradache Mult-attrbute Decso Makg based o Rough eutrosophc Varato Coeffcet Smlarty Measure X X = are the Eucldea orm of X ad Y XY = y s the er product of the vector X ad Y Proposto 5 [83] Jaccard de satsfes the followg propertes: 0 JX Y

7 eutrosophc Sets ad Systems Vol JX Y = JY X 3 JX Y = for X = Y e = y = for every X ad y Y Defto 3 [84] he Dce smlarty measure ca be defed as follows: X Y y EX Y = = X + Y y Proposto 6 [84] he Dce smlarty measure satsfes the followg propertes: 0 EX Y EX Y = EY X 3 JX Y = for X = Y e = y = for every X ad y Y Defto 33 [85] he cose smlarty measure betwee two vectors X ad Y s the er product of these two vectors dvded by the product of ther legths ad ca be defed as follows: X Y y CX Y = = 3 X Y y Proposto 7 [85] he cose smlarty measure satsfes the followg propertes 0 CX Y CX Y = CY X 3 CX Y = for X = Y e = y = for every X ad y Y hese three formulas are smlar the sese that they take values the terval [0 ] Jaccard ad Dce smlarty measures are udefed whe = 0 ad y = 0 for = ad cose smlarty measure s udefed whe = 0 or y = 0 for = Defto 34 [86] Varatoal co-effcet smlarty measure ca be defed as follows: X Y X Y VX Y = λ + - X + Y X Y y y y + 4 y Proposto 8 [86] Varatoal co-effcet smlarty measure satsfes the followg propertes: 0 VX Y VX Y = VY X 3 VX Y = for X = Y e = y = for every X ad y Y 4 Varous smlarty measures for sgle valued eutrosophc sets ssume ad be two SVSs a uverse of dscourse X = [0] for ay X or [0] for ay X ca be cosdered as a vector represetato wth three elemets Let w [0] be the weght of each elemet for = such that w the Jaccard Dce ad cose smlarty measures ca be preseted as follows: Defto 4[0] Jaccard smlarty measure betwee ad ca be defed as follows: Jac = { } Proposto 9 [0] Jaccard smlarty measure satefes the followg propertes: 0 Jac ; Jac Jac ; 3 Jac ; f = e ad for every = X Defto 4 [0] Weghted Jaccard smlarty measure betwee ad ca be defed as follows: Jac w = w { } Proposto 0 [0] Weghted Jaccard smlarty measure satsfes the followg propertes: 0Jac w ; Jac w Jac w ; 3 Jac w ; f = e ad for every = X 5 6 Kalya Modal Surapat Pramak ad loret Smaradache Mult-attrbute Decso Makg based o Rough eutrosophc Varato Coeffcet Smlarty Measure

8 eutrosophc Sets ad Systems Vol 3 06 Kalya Modal Surapat Pramak ad loret Smaradache Mult-attrbute Decso Makg based o Rough eutrosophc Varato Coeffcet Smlarty Measure Defto 4 [] Dce smlarty measure betwee ad s defed as: Dc = 7 Proposto [] Dce smlarty measure satsfes the followg propertes: ; 0 Dc Dc ; Dc ; 3 Dc f = e ad for every = X Defto 4 [] Weghted Dce smlarty measure betwee ad ca be defed as follows: Dc w = w 8 Proposto [] Weghted Dce smlarty measure ; 0 Dc w ; Dc Dc w w ; 3 Dc w f = e ad for every = X Defto 43 [] Cose smlarty measure betwee ad ca be defed as follows: Cos = 9 Proposto 3 [] Cose smlarty measure satsfes the followg propertes: ; 0 Cos w Cos Cos w w ; 3 Cos w f = e ad for every = X Defto 43 [] Weghted cose smlarty measure betwee ad ca be defed as follows: Cos w = w 0 Proposto 4 [] Weghted cose smlarty measure satsfes the followg propertes: ; 0 Cos w Cos Cos w w ; 3 Cos w f = e ad for every = X Jaccard ad Dce smlarty measures betwee two eutrosophc sets ad are udefed whe 0 ad 0 for all = Smlarly the cose formula for two eutrosophc sets ad s udefed whe 0 or 0 for all = 5 Varatoal smlarty measures for rough eutrosophc sets he oto of rough eutrosophc set RS s used as vector represetatos 3D-vector space ssume that X = ad Y = y y y be two -dmesoal vectors wth postve co-ordates Jaccard Dce cose ad cotaget smlarty measures betwee two vectors are stated as follows Defto 5 [] Jaccard smlarty measure uder rough eutrosophc evromet ssume that ad X = be ay two rough eutrosophc sets Jacard smlarty measure [] betwee rough eutrosophc sets ad ca be defed as follows: Jac RS = 7

9 eutrosophc Sets ad Systems Vol 3 06 Kalya Modal Surapat Pramak ad loret Smaradache Mult-attrbute Decso Makg based o Rough eutrosophc Varato Coeffcet Smlarty Measure ] [ Proposto 5 [] Jaccard smlarty measure [] betwee ad satsfes the followg propertes: ; 0 Jac RS ; Jac Jac RS RS ; 3 Jac RS ff = 4 f C s a RS Y ad C the Jac RS C Jac RS ad Jac RS C Jac RS C Defto 5 [] f we cosder the weghts of each elemet weghted rough Jaccard smlarty measure [] betwee rough eutrosophc sets ad ca be defed as follows: Jac WRS = w ] [ [0] w = ad w f we take w = the Jac WRS = Jac RS Proposto 6 [] he weghted rough Jaccard smlarty [] measure betwee two rough eutrosophc sets ad also satsfes the followg propertes: ; 0 Jac WRS ; Jac Jac WRS WRS ; 3 Jac WRS ff = 4 f C s a WRS Y ad C the Jac WRS C Jac WRS ad Jac WRS C Jac WRS C Defto 5 [] Dce smlarty measure uder rough eutrosophc evromet ths secto Dce smlarty measure ad the weghted Dce smlarty measure for rough eutrosophc sets have bee stated due to Pramak ad Modal [] Suppose that ad be ay two rough eutrosophc sets X = Dce smlarty measure betwee rough eutrosophc sets ad ca be defed as follows: DC RS = 3 Proposto 7 [] Dce smlarty measure [] satsfes the followg propertes ; 0 DC RS ; DC DC RS RS ; 3 DC RS ff = 4 f C s a RS Y ad C the DC RS C DC RS ad DC RS C DC RS C or proofs of the above metoed four propertes see [] Defto 5 f we cosder the weghts of each elemet a weghted rough Dce smlarty measure betwee rough eutrosophc sets ad ca be defed as follows: DC WRS = w 4 [0] w = ad w f we take w = the DC WRS = DC RS Proposto 8 [] he weghted rough Dce smlarty [] measure betwee two rough eutrosophc sets ad also satsfes the followg propertes: ; 0 DC WRS ; DC DC WRS WRS ; 3 DC WRS ff = 4 f C s a RS Y ad C the DC WRS C DC WRS ad DC WRS C DC WRS C or proofs of the above metoed four propertes see [] Defto 53 [0] Cose smlarty measure ca be defed as the er product of two vectors dvded by the product of ther legths t s the cose of the agle betwee the vector represetatos of two rough eutrosophc sets he cose smlarty measure s a fudametal measure used formato techology Pramak ad Modal [0] 8

10 eutrosophc Sets ad Systems Vol 3 06 Kalya Modal Surapat Pramak ad loret Smaradache Mult-attrbute Decso Makg based o Rough eutrosophc Varato Coeffcet Smlarty Measure defed cose smlarty measure betwee rough eutrosophc sets 3-D vector space ssume that ad X = be ay rough eutrosophc sets Pramak ad Modal [0] defed cose smlarty measure betwee rough eutrosophc sets ad as follows: C RS = 5 Here Proposto 9 [0] Let ad be rough eutrosophc sets Cose smlarty measure [0] betwee ad satsfes the followg propertes ; 0 C RS ; C C RS RS ; 3 C RS ff = 4 f C s a RS Y ad C the C RS C C RS ad C RS C C RS C Defto 53 [0] f we cosder the weghts of each elemet a weghted rough cose smlarty measure betwee rough eutrosophc sets ad ca be defed as follows: C WRS = w 6 [0] w = ad w f we take w = the C WRS = C RS Proposto 0 [0] he weghted rough cose smlarty measure [0] betwee two rough eutrosophc sets ad also satsfes the followg propertes: ; 0 C WRS ; C C WRS WRS ; 3 C WRS ff = 4 f C s a WRS Y ad C the C WRS C C WRS ad C WRS C C WRS C or proofs of the above metoed four propertes see [0] Defto 54 [9] Cotaget smlarty measures of rough eutrosophc sets ssume that ad X = be ay two rough eutrosophc sets Pramak ad Modal [9] defed cotaget smlarty measure betwee rough eutrosophc sets ad as follows: CO RS = 3 cot 7 Here Proposto [9] Cotaget smlarty measure satsfes the followg propertes: ; 0 CO RS ; CO CO RS RS ; 3 CO RS ff = 4 f C s a RS Y ad C the CO RS C CO RS ad CO RS C CO RS C Defto 54 f we cosder the weghts of each elemet a weghted rough cotaget smlarty measure [9] betwee rough eutrosophc sets ad ca be defed as follows: CO WRS = w 3 cot 8 [0] w = ad w f we take w = the CO WRS = CO RS Proposto [9] he weghted rough cose smlarty measure betwee two rough eutrosophc sets ad also satsfes the followg propertes: ; 0 CO WRS ; CO CO WRS WRS ; 3 CO WRS ff = 4 f C s a WRS Y ad C the CO WRS C CO WRS ad CO WRS C CO WRS C 9

11 eutrosophc Sets ad Systems Vol 3 06 Kalya Modal Surapat Pramak ad loret Smaradache Mult-attrbute Decso Makg based o Rough eutrosophc Varato Coeffcet Smlarty Measure Defto 55 Varatoal co-effcet smlarty measure betwee rough eutrosophc sets Let ad be two rough eutrosophc sets Varatoal co-effcet smlarty measure betwee rough eutrosophc sets ca be preseted as follows: Var RS = 9 Here Proposto 3 he varatoal co-effcet smlarty measure Var RS betwee two rough eutrosophc sets ad satsfes the followg propertes: ; 0 Var RS ; Var Var RS RS ; 3 Var RS f = e ad for every = X Proof t s obvous that 0 Var RS hus t s requred to prove that Var RS rom rough eutrosophc dce smlarty measure t ca be wtte that 0 0 ad from rough eutrosophc cose smlarty measure t ca be wrtte that 0 Combg Eq0 ad Eq we obta Var RS = hus ; 0 Var RS Var RS = = Var RS 3 f = e ad for every = X Var RS = 0

12 eutrosophc Sets ad Systems Vol 3 06 Kalya Modal Surapat Pramak ad loret Smaradache Mult -attrbute Decso Makg based o Rough eutrosophc Varato Coeffcet Smlarty Measure = hese results show the completo of the proofs of the three propertes Defto 56 Weghted varatoal co-effcet smlarty measure betwee rough eutrosophc sets Let ad be ay two rough eutrosophc sets Rough varatoal co-effcet smlarty measure betwee rough eutrosophc sets ad 3-D vector space ca be preseted as follows: Var WRS = w w 3 f w the Eq3 s reduced to Eq9 Proposto 4 he weghted varatoal co-effcet smlarty measure also satsfes the followg propertes: ; 0 Var WRS ; Var Var WRS WRS 3 Var WRS = ; f = e ad for every = X Proof: t s obvous that 0 Var W RS hus t s requred to prove that Var WRS rom rough eutrosophc weghted dce smlarty measure t ca be wrtte that 0 w 4 ad from rough eutrosophc weghted cose smlarty measure t ca be wrtte that 0 w 5 Combg Eq4 ad Eq5 we obta Var WRS = w w 6 hus ; 0 Var WRS Var WRS = w w = w w Var WRS 3 f = e

13 eutrosophc Sets ad Systems Vol 3 06 ad for every = X Var WRS = w w = w w hese results show the completo of the proofs of the three propertes 6 Mult attrbute decso makg based o rough eutrosophc varatoal coeffcet smlarty measure ths secto a rough varatoal co-effcet smlarty measure s employed to mult-attrbute decso makg rough eutrosophc evromet ssume that = { m } be the set of alteratves ad C = {C C C } be the set of attrbutes a mult-attrbute decso makg problem ssue that w j be the weght of the attrbute C j provded by the decso maker such that each w [0] ad j w j However real stuato decso maker may ofte face dffculty to evaluate alteratves over the attrbutes due to vague or complete formato about alteratves a decso makg stuato Rough eutrosophc set ca be used MDM to deal wth complete formato of the alteratves ths paper the assessmet values of all the alteratves wth respect to attrbutes are cosdered as the rough eutrosophc values see able able: Rough eutrosophc decso matr DRS d j d j m = m Here C d d d d m j d d d d m j C d d d d m d d m -th alteratve ad the j-th attrbute C d d d d m d d m 7 s the rough eutrosophc umber for the Defto 6: rasformg operator for SVSs [80] he rough eutrosophc decso matr 7 ca be trasformed to sgle valued eutrosophc decso matr whose j-th elemet ca be preseted as follows: djdj j m for = 3 m; j j = 3 8 Step Determe the eutrosophc relatve postve deal soluto mult-crtera decso-makg evromet the cocept of deal pot has bee used to help detfy the best alteratve the decso set Defto 6 [5] Let H be the collecto of two types of attrbutes amely beeft type attrbute P ad cost type attrbute L the MDM problems he relatve postve deal eutrosophc soluto RPS Q ] s the soluto of the decso matr S [ q S q S q S DS j j j m every compoet of Q S has the followg form: where for beeft type attrbute every compoet of Q S followg form: q S j j j ma{ j}m{ j}m{ j} for j P ad for cost type attrbute every compoet of Q S the followg form q j j j S m{ }ma{ }ma{ } j j j for j L has the 9 has 30 Step Determe the weghted varatoal co-effcet smlarty measure betwee deal alteratve ad each alteratve he varatoal co-effcet smlarty measure betwee deal alteratve Q S ad each alteratve for = m ca be determed by the followg equato as follows: Var WRS QS DS j j j j j j j j j j j j j j j j j j w j j j w 3 j j j Step3 Rak the alteratves ccordg to the values obtaed from Eq3 the rakg order of all the alteratves ca be easly determed Hghest value dcates the best alteratve Kalya Modal Surapat Pramak ad loret Smaradache Mult-attrbute Decso Makg based o Rough eutrosophc Varato Coeffcet Smlarty Measure

14 eutrosophc Sets ad Systems Vol Step 4 Ed 7 umercal eample ths secto rough eutrosophc MDM regardg vestmet problem s cosdered to demostrate the applcablty ad the effectveess of the proposed approach However vestmet problem s ot easy to solve t ot oly requres oodles of patece ad dscple but also a great deal of research ad a soud uderstadg of the market mathematcal tools amog others Suppose a vestmet compay wats to vest a sum of moey the best opto ssume that there are four possble alteratves to vest the moey: s a computer compay; s a garmet compay; 3 3 s a telecommucato compay; ad 4 4 s a food compay he vestmet compay must take a decso based o the followg three crtera: C s the growth factor; C s the evrometal mpact; ad 3 C 3 s the rsk factor he four possble alteratves are to be evaluated uder the attrbute by the rough eutrosophc assessmets provded by the decso maker hese assessmet values are gve the rough eutrosophc decso matr see the table able Rough eutrosophc decso matr D j P P j C C C he kow weght formato s gve as follows: W = [w w w 3 ] 3 = [ ] ad w 3 Step Determe the types of crtera rst two types e C ad C of the gve crtera are beeft type crtera ad the last oe crtero e C 3 s the cost type crtera Step Determe the relatve eutrosophc postve deal soluto Usg Eq 9 Eq30 the relatve postve deal eutrosophc soluto for the gve matr defed Eq3 ca be obtaed as: Q S [ ] Step3 Determe the weghted varatoal smlarty measure he weghted varatoal co-effcet smlarty measure s determed by usg Eq8 Eq3 ad Eq3 he results obtaed for dfferet values of have bee show the able-3 able-3 Results of rough varatoal smlarty measure for dfferet values of 0 Smlarty measure method Values of s Measure values Rakg order ; 0974; 0997; > > > 4 VarWRS QS DS > > > ; 09735; 09887; > > > ; 09730; 09868; > > > ; 0978; 09857; > > > 4 Step 4 Rak the alteratves ccordg to the dfferet values of the results obtaed able-3 reflects that 3 s the best alteratve 8 Comparsos of dfferet rough smlarty measure wth rough varato smlarty measure ths secto four estg rough smlarty measures - amely: rough cose smlarty measure rough dce smlarty measure rough cotaget smlarty measure ad rough Jaccard smlarty measure - have bee compared wth proposed rough varatoal co-effcet smlarty measure for dfferet values of he comparso results are lsted the able 3 ad able 4 Kalya Modal Surapat Pramak ad loret Smaradache Mult-attrbute Decso Makg based o Rough eutrosophc Varato Coeffcet Smlarty Measure

15 eutrosophc Sets ad Systems Vol able-4 Results of estg rough eutrosophc smlarty measure methods Rough smlarty measure methods Values of s Measure values Rakg order JCWRS QS DS DCWRS QS DS CWRS QS DS COWRS QS DS [] ; 09739; > > > 4 [] 08595; 0976; 09873; > > > 4 [0] 08788; 09738; 0990; > > 4 > [9] 0847; 09358; 09643; > > > 4 Cocluso ths paper we have proposed rough varatoal coeffcet smlarty measures We also proved some of ther basc propertes We have preseted a applcato of rough eutrosophc varatoal coeffcet smlarty measure for a decso makg problem o vestmet he cocept preseted the paper ca be appled to deal wth other mult attrbute decso makg problems rough eutrosophc evromet Refereces [] L Zadeh uzzy sets formato ad Cotrol [] K taassov tutostc fuzzy sets uzzy Sets ad Systems [3] Smaradache ufyg feld logcs eutrosophy: eutrosophc probablty set ad logc merca Research Press Rehoboth 998 [4] Smaradache eutrosophc set- a geeralzato of tutostc fuzzy sets teratoal Joural of Pure ad ppled Mathematcs [5] Smaradache eutrosophc set-a geeralzato of tutostc fuzzy set Joural of Defese Resources Maagemet [6] H Wag Smaradache Y Q Zhag ad R Suderrama Sgle valued eutrosophc sets Multspace ad Multstructure [7] Z Pawlak Rough sets teratoal Joural of formato ad Computer Sceces [8] S roum Smaradache ad M Dhar Rough eutrosophc sets tala Joural of Pure ad ppled Mathematcs [9] S roum Smaradache ad M Dhar Rough eutrosophc sets eutrosophc Sets ad Systems [0] J Ye Vector smlarty measures of smplfed eutrosophc sets ad ther applcato mult-crtera decso makg teratoal Joural of uzzy Systems [] S Ye ad J Ye Dce smlarty measure betwee sgle valued eutrosophc mult-sets ad ts applcato medcal dagoss eutrosophc Sets ad Systems [] J Ye mproved cose smlarty measures of smplfed eutrosophc sets for medcal dagoses rtfcal tellgece ad Medce 04 do: 006/jartmed04007 [3] J Ye Multple attrbute group decso-makg method wth completely ukow weghts based o smlarty measures uder sgle valued eutrosophc evromet Joural of tellgece ad uzzy Systems 04 do: 0333/S-45 [4] J Ye ad Q S Zhag Sgle valued eutrosophc smlarty measures for multple attrbute decso makg eutrosophc Sets ad Systems [5] J Ye Clusterg methods usg dstace-based smlarty measures of sgle-valued eutrosophc sets Joural of tellgece Systems 04 do: 055/jsys [6] P swas S Pramak ad C Gr Cose smlarty measure based mult-attrbute decso-makg wth trapezodal fuzzy eutrosophc umbers eutrosophc Sets ad Systems [7] K Modal ad S Pramak eutrosophc refed smlarty measure based o taget fucto ad ts applcato to mult attrbute decso makg Joural of ew heory [8] K Modal ad S Pramak eutrosophc refed smlarty measure based o cotaget fucto ad ts applcato to mult attrbute decso makg Global Joural of dvaced Research [9] S Pramak ad K Modal Cotaget smlarty measure of rough eutrosophc sets ad ts applcato to medcal dagoss Joural of ew heory [0] S Pramak ad K Modal Cose smlarty measure of rough eutrosophc sets ad ts applcato medcal dagoss Global Joural of dvaced Research 05-0 [] S Pramak ad K Modal Some rough eutrosophc smlarty measure ad ther applcato to mult attrbute decso makg Global Joural of Egeerg Scece ad Research Maagemet [] K Modal ad S Pramak r-comple rough eutrosophc smlarty measure ad ts applcato mult-attrbute decso makg Crtcal Revew [3] K Modal S Pramak ad Smaradache Hypercomple rough eutrosophc smlarty measure ad ts applcato mult-attrbute decso makg Crtcal Revew 3 07 Press [4] Modal S Pramak ad Smaradache Several trgoometrc Hammg smlarty measures of rough eutrosophc sets ad ther applcatos decso makg Smaradache & S Pramak Eds ew treds eutro- Kalya Modal Surapat Pramak ad loret Smaradache Mult-attrbute Decso Makg based o Rough eutrosophc Varato Coeffcet Smlarty Measure

16 eutrosophc Sets ad Systems Vol sophc theory ad applcatos russells Pos Edtos [5] L Hwag ad K Yoo Multple attrbute decso makg: methods ad applcatos Sprger ew York 98 [6] C L Hwag ad K Yoo Multple attrbute decso makg: methods ad applcatos State of the rt Survey Sprger-Verlag erl 98 [7] M Ehrgott ad X Gadbbleu Multple crtera optmzato: state of the art aotated bblography survey Kluwer cademc Publshers osto 00 [8] L Hwag ad M J L Group decso makg uder multple crtera: methods ad applcatos SprgerVerlag Hedelberg 987 [9] R R Yager O ordered weghted averagg aggregato operators multcrtera decso makg EEE rasactos o Systems Ma ad Cyberetcs [30] Kaya ad C Kahrama Mult crtera decso makg eergy plag usg a modfed fuzzy OPSS methodology Epert System wth pplcatos [3] M Mergó ad M Gl-Lafuete uzzy duced geeralzed aggregato operators ad ts applcato multperso decso makg Epert Systems wth pplcatos [3] L L X H Yua ad Z Q Xa Multcrtera fuzzy decso-makg based o tutostc fuzzy sets Joural of Computers ad Systems Sceces [33] H W Lu ad G J Wag Mult-crtera decso makg methods based o tutostc fuzzy sets Europea Joural of Operatoal Research [34] Z S Xu ad R R Yager Dyamc tutostc fuzzy mult-attrbute decso makg teratoal Joural of ppromate Reasog [35] E oraa S Geça M Kurtb ad D kay multcrtera tutostc fuzzy group decso makg for suppler selecto wth OPSS method Epert Systems wth pplcatos [36] S Pramak ad D Mukhopadhyaya Grey relatoal aalyss based tutostc fuzzy mult crtera group decsomakg approach for teacher selecto hgher educato teratoal Joural of Computer pplcatos [37] K Modal ad S Pramak tutostc fuzzy multcrtera group decso makg approach to qualtybrck selecto problem Joural of ppled Quattatve Methods [38] S P Wa ad J Y Dog possblty degree method for terval-valued tutostc fuzzy mult-attrbute group decso makg Joural of Computer ad System Sceces [39] K Modal ad S Pramak tutostc fuzzy smlarty measure based o taget fucto ad ts applcato to mult-attrbute decso makg Global Joural of dvaced Research [40] P P Dey S Pramak ad C Gr Mult-crtera group decso makg tutostc fuzzy evromet based o grey relatoal aalyss for weaver selecto Khad sttuto Joural of ppled ad Quattatve Methods [4] J Ye Multcrtera decso-makg method usg the correlato coeffcet uder sgle-valued eutrosophc evromet teratoal Joural of Geeral Systems [4] J Ye Sgle valued eutrosophc cross-etropy for multcrtera decso makg problems ppled Mathematcal Modellg [43] J J Peg J Q Wag H Y Zhag ad X H Che outrakg approach for mult-crtera decso-makg problems wth smplfed eutrosophc sets ppled Soft Computg [44] Kharal eutrosophc mult-crtera decso makg method ew Mathematcs ad atural Computato [45] P swas S Pramak ad C Gr Etropy based grey relatoal aalyss method for mult-attrbute decso makg uder sgle valued eutrosophc assessmets eutrosophc Sets ad Systems [46] P swas S Pramak ad C Gr ew methodology for eutrosophc mult-attrbute decso makg wth ukow weght formato eutrosophc Sets ad Systems [47] J Ye multcrtera decso-makg method usg aggregato operators for smplfed eutrosophc sets Joural of tellget ad uzzy Systems [48] K Modal ad S Pramak Mult-crtera group decso makg approach for teacher recrutmet hgher educato uder smplfed eutrosophc evromet eutrosophc Sets ad Systems [49] J J Peg J Q Wag J Wag H Y Zhag ad X H Che Smplfed eutrosophc sets ad ther applcatos mult-crtera group decso-makg problems teratoal Joural of Systems Scece [50] R Sah ad P Lu Mamzg devato method for eutrosophc multple attrbute decso makg wth complete weght formato eural Computg ad pplcatos [5] P swas S Pramak ad C Gr OPSS method for mult-attrbute group decso-makg uder sgle-valued eutrosophc evromet eural Computg ad pplcatos [5] J Ye rapezodal eutrosophc set ad ts applcato to multple attrbute decso-makg eural Computg ad pplcatos [53] J Ye drectoal projecto method for multple attrbute group decso makg wth eutrosophc umbers eural Computg ad pplcatos 05 do: 0007/s [54] S Pramak S Dalapat ad K Roy Logstcs ceter locato selecto approach based o eutrosophc multcrtera decso makg Smaradache & S Pramak Eds ew treds eutrosophc theory ad applcatos russells Pos Edtos [55] S Pramak D aerjee ad C Gr Mult crtera group decso makg model eutrosophc refed set ad ts applcato Global Joural of Egeerg Scece ad Research Maagemet Kalya Modal Surapat Pramak ad loret Smaradache Mult-attrbute Decso Makg based o Rough eutrosophc Varato Coeffcet Smlarty Measure

17 eutrosophc Sets ad Systems Vol [56] P swas S Pramak ad C Gr ggregato of tragular fuzzy eutrosophc set formato ad ts applcato to mult-attrbute decso makg eutrosophc Sets ad Systems [57] P swas S Pramak ad C Gr Value ad ambguty de based rakg method of sgle-valued trapezodal eutrosophc umbers ad ts applcato to mult-attrbute decso makg eutrosophc Sets ad Systems [58] Z a J Wag J Wag ad H Zhag Smplfed eutrosophc lgustc mult-crtera group decso-makg approach to gree product developmet Group Decso ad egotato 06 do: 0007/s [59] Z a J Wag J Wag ad H Zhag mproved MULMOOR approach for mult-crtera decso makg based o terdepedet puts of smplfed eutrosophc lgustc formato eural Computg ad pplcatos [60] P rmal ad M G hatt Selecto of materal hadlg automated guded vehcle usg fuzzy sgle valued eutrosophc set-etropy based ovel mult attrbute decso makg techque mplemetato ad valdato Smaradache & S Pramak Eds ew treds eutrosophc theory ad applcatos russells Pos Edtos [6] J Q Wag Y Yag ad L L Mult-crtera decsomakg method based o sgle-valued eutrosophc lgustc Maclaur symmetrc mea operators eural Computg ad pplcatos Do:0007/s [6] K Madal ad K asu Mult crtera decso makg method eutrosophc evromet usg a ew aggregato operator score ad certaty fucto Smaradache & S Pramak Eds ew treds eutrosophc theory ad applcatos russells Pos Edtos [63] PP Dey S Pramak ad C Gr eteded grey relatoal aalyss based multple attrbute decso makg terval eutrosophc ucerta lgustc settg eutrosophc Sets ad Systems [64] P Ch ad P Lu eteded OPSS method for the mult-attrbute decso makg problems o terval eutrosophc set eutrosophc Sets ad Systems [65] H Zhag P J J Wag ad X Che mproved weghted correlato coeffcet based o tegrated weght for terval eutrosophc sets ad ts applcato multcrtera decso makg problems teratoal Joural of Computatoal tellgece Systems [66] S roum J Ye d Smaradache eteded OP- SS method for multple attrbute decso makg based o terval eutrosophc ucerta lgustc varables eutrosophc Sets ad Systems [67] P P Dey S Pramak ad C Gr Eteded projectobased models for solvg multple attrbute decso makg problems wth terval valued eutrosophc formato Smaradache & S Pramak Eds ew treds eutrosophc theory ad applcatos russells Pos Edtos [68] S Pramak ad K Modal terval eutrosophc multattrbute decso-makg based o grey relatoal aalyss eutrosophc Sets ad Systems [69] S Pramak P P Dey ad C Gr OPSS for sgle valued eutrosophc soft epert set based mult-attrbute decso makg problems eutrosophc Sets ad Systems [70] Del M l ad Smaradache polar eutrosophc sets ad ther applcato based o mult-crtera decso makg Proceedgs of the 05 teratoal Coferece o dvaced Mechatroc Systems ejg Cha [7] P P Dey S Pramak ad C Gr OPSS for solvg mult-attrbute decso makg problems uder b-polar eutrosophc evromet Smaradache & S Pramak Eds ew treds eutrosophc theory ad applcatos russells Pos Edtos [7] P P Dey S Pramak ad C Gr Geeralzed eutrosophc soft mult-attrbute group decso makg based o OPSS Crtcal Revew [73] P P Dey S Pramak ad C Gr eutrosophc soft mult-attrbute decso makg based o grey relatoal projecto method eutrosophc Sets ad Systems [74] P P Dey S Pramak ad C Gr eutrosophc soft mult-attrbute group decso makg based o grey relatoal aalyss method Joural of ew Results Scece [75] P K Maj eutrosophc soft set als of uzzy Mathematcs ad formatc [76] P K Maj Weghted eutrosophc soft sets approach a mult-crtera decso makg problem Joural of ew heory [77] P swas S Pramak ad C Gr Some dstace measures of sgle valued eutrosophc hestat fuzzy sets ad ther applcatos to multple attrbute decso makg Smaradache & S Pramak Eds ew treds eutrosophc theory ad applcatos russells Pos Edtos [78] P swas S Pramak ad C Gr GR method of multple attrbute decso makg wth sgle valued eutrosophc hestat fuzzy set formato Smaradache & S Pramak Eds ew treds eutrosophc theory ad applcatos russells Pos Edtos [79] R Sah ad P Lu Dstace ad smlarty measures for multple attrbute decso makg wth sgle valued eutrosophc hestat fuzzy formato Smaradache & S Pramak Eds ew treds eutrosophc theory ad applcatos russells Pos Edtos [80] K Modal ad S Pramak Rough eutrosophc multattrbute decso-makg based o rough accuracy score fucto eutrosophc Sets ad Systems [8] K Modal ad S Pramak Rough eutrosophc multattrbute decso-makg based o grey relatoal aalyss eutrosophc Sets ad Systems Kalya Modal Surapat Pramak ad loret Smaradache Mult-attrbute Decso Makg based o Rough eutrosophc Varato Coeffcet Smlarty Measure

18 eutrosophc Sets ad Systems Vol [8] S Pramak P swas ad C Gr Hybrd vector smlarty measures ad ther applcatos to mult-attrbute decso makg uder eutrosophc evromet eural Computg ad pplcatos 05 do:0007/s [83] P Jaccard Dstrbuto de la flore alpe das le ass des quelques regos voses ull de la Socette Vaudose des Sceces aturelles [84] L R Dce Measures of amout of ecologc assocato betwee speces Ecology [85] G Salto ad M J McGll troducto to moder formato retreval ucklad McGraw-Hll 983 [86] X Xu L Zhag ad Q Wa varatoal coeffcet smlarty measure ad ts applcato emergecy group decso makg System Egeerg Proceda Receved: ovember 0 06 ccepted: December 5 06 Kalya Modal Surapat Pramak ad loret Smaradache Mult-attrbute Decso Makg based o Rough eutrosophc Varato Coeffcet Smlarty Measure

19 eutrosophc Sets ad Systems Vol Uversty of ew Meco Regular Sgle Valued eutrosophc Hypergraphs Muhammad slam Malk l Hassa Sad roum 3 ad Smaradache 4 Departmet of mathematcs Uversty of Pujab Lahore Paksta E-mal: aslam@mathpuedupk malkpu@yahoocom Departmet of mathematcs Uversty of Pujab Lahore Paksta E-mal: alhassaumath@gmalcom 3 Laboratory of formato Processg aculty of Scece e M Sk Uversty Hassa P 7955 Sd Othma Casablaca Morocco 4 Uversty of ew Meco Mathematcs & Scece Departmet 705 Gurley ve Gallup M 8730 US E-mal: fsmaradache@gmalcom bstract ths paper we defe the regular ad totally regular sgle valued eutrosophc hypergraphs ad dscuss the order ad sze alog wth propertes of regular ad totally regular sgle valued eutrosophc hypergraphs We also eted work o completeess of sgle valued eutrosophc hypergraphs Keywords: Sgle valued eutrosophc hypergraphs regular sgle valued eutrosophc hypergraphs ad totally regular sgle valued eutrosophc hypergraphs troducto he oto of eutrosophc sets Ss was proposed by Smaradache [8] as a geeralzato of the fuzzy sets [4] tutostc fuzzy sets [] terval valued fuzzy set [] ad terval-valued tutostc fuzzy sets [3] theores he eutrosophc set s a powerful mathematcal tool for dealg wth complete determate ad cosstet formato real world he eutrosophc sets are characterzed by a truth-membershp fucto t a determacy-membershp fucto ad a falsty membershp fucto f depedetly whch are wth the real stadard or ostadard ut terval ] [ order to coveetly use S real lfe applcatos Wag et al [9] troduced the cocept of the sgle-valued eutrosophc set SVS a subclass of the eutrosophc sets he same authors [0] troduced the cocept of the terval valued eutrosophc set VS whch s more precse ad fleble tha the sgle valued eutrosophc set he VS s a geeralzato of the sgle valued eutrosophc set whch the three membershp fuctos are depedet ad ther value belog to the ut terval [0 ] More works o sgle valued eutrosophc sets terval valued eutrosophc sets ad ther applcatos ca be foud o Hypergraph s a graph whch a edge ca coect more tha two vertces hypergraphs ca be appled to aalyse archtecture structures ad to represet system parttos Mordese J ad PS asr gave the deftos for fuzzy hypergraphs Parvathy R ad M G Karuambga s paper troduced the cocepts of tutostc fuzzy hypergraphs ad aalyse ts compoets agoor Ga ad Sajth egum S defed degree order ad sze tutostc fuzzy graphs ad eted the propertes agoor Ga ad Latha R troduced rregular fuzzy graphs ad dscussed some of ts propertes Regular tutostc fuzzy hypergraphs ad totally regular tutostc fuzzy hypergraphs are troduced by Pradeepa ad Vmala S [0] ths paper we eted regularty ad totally regularty o sgle valued eutrosophc hypergraphs Prelmares Defto Let X be a space of pots objects wth geerc elemets X deoted by sgle valued eutrosophc set SVS s characterzed by truth membershp fucto determacy membershp fucto ad a falsty membershp fucto or each pot X; [0 ] Defto Let be a SVS o X the support of s deoted ad defed by Supp = { : X > 0 > 0 > 0 } Muhammad slam Malk l Hassa Sad roum loret Smaradace Regular Sgle Valued eutrosophc Hypergraphs

20 eutrosophc Sets ad Systems Vol Defto 3 hypergraph s a ordered par H = X E where X = { } be a fte set of vertces E = {E E E m } be a famly of subsets of X 3 E j for j = 3m ad E j j = X he set X s called set of vertces ad E s the set of edgesor hyper edges Defto 4 he sgle valued eutrosophc hypergraph s a ordered par H = X E where X = { } be a fte set of vertces E = {E E E m } be a famly of SVSs of X 3 E j O = for j = 3 m ad j SuppE j = X he set X s called set of vertces ad E s the set of SV-edgesor SV-hyperedges Proposto 5 he sgle valued eutrosophc hypergraph s the geeralzato of fuzzy hypergraphs ad tutostc fuzzy hypergraphs 3 Regular ad totally regular SVHGs Defto 3 he ope eghbourrhood of a verte sgle valued eutrosophc hypergraphs SVHGs s the set of adjacet vertces of ecludg that verte ad s deoted by Defto 3 he closed eghbourhood of a verte sgle valued eutrosophc hypergraphs SVHGs s the set of adjacet vertces of cludg that verte ad s deoted by [] Eample 33 Cosder a sgle valued eutrosophc hyper graphs H = X E where X = {a b c d e} ad E = {P Q R S} whch are defed by P = {a 3 b 4 5 6} Q = {c 3 d e 7 8 9} R = {b 3 c 4 5 6} S = {a 3 d 4 5 6} he the ope eghbourhood of a verte a cota b ad d he closed eghbourhood of a verte b cota b a ad c Defto 34 Let H = X E be a SVHG the ope eghbourhood degree of a verte whch s deoted ad defed by deg = deg deg deg where deg = deg = deg = X X X E E E Eample 35 Cosder a sgle valued eutrosophc hypergraphs H = X E where X = {a b c d e} ad E = {P Q R S} whch are defed by P = {a 3 b 4 5 6} Q = {c 3 d e 7 8 9} R = {b 3 c 4 5 6} S = {a 3 d 4 5 6} he the ope eghbourhood of a verte a s b ad d ad therefore the ope eghbourhood degree degree of a verte a s 8 Defto 36 Let H = X E be a SVHG the closed eghbourhood degree of a verte whch s deoted ad defed by where deg[] = deg [] deg [] deg [] deg [] = deg + E deg [] = deg + E deg [] = deg + E Eample 37 Cosder a sgle valued eutrosophc hypergraphs H = X E where X = {a b c d e} ad E = {P Q R S} whch s defed by P = {a 3 b 4 5 6} Q = {c 3 d e 7 8 9} Muhammad slam Malk l Hassa Sad roum loret Smaradace Regular Sgle Valued eutrosophc Hypergraphs

21 eutrosophc Sets ad Systems Vol R = {b 3 c 4 5 6} S = {a 3 d 4 5 6} he closed eghbourhood of a verte b cota b a ad c hece the closed eghbourhood degree of a verte a s 9 5 Defto 38 Let H = X E be a SVHG the H s sad to be a -regular SVHG f all the vertces have the same ope eghbourhood degree = 3 Defto 39 Let H = X E be a SVHG the H s sad to be a m-totally regular SVHG f all the vertces have the same closed eghbourhood degree m = m m m 3 Proposto 30 regular SVHG s the geeralzato of regular fuzzy hypergraphs ad regular tutostc fuzzy hypergraphs Proposto 3 totally regular SVHG s the geeralzato of totally regular fuzzy hypergraphs ad totally regular tutostc fuzzy hypergraphs Eample 3 Cosder a sgle valued eutrosophc hypergraphs H = X E where X = {a b c d} ad E = {P Q R S} whch are defed by P = {a 8 3 b 8 3} Q = {b 8 3 c 8 3} R = {c 8 3 d 8 3} S = {d 8 3 a 8 3} Here the ope eghbourhood degree of every verte s hece H s regular SVHG ad closed eghbourhood degree of every verte s Hece H s both regular ad totally regular SVHG heorem 33 Let H = X E be a SVHG whch s both regular ad totally regular SVHG the E s costat Proof: Suppose H s a -regular ad a m-totally regular SVHG he deg = = 3 deg[] = m = m m m 3 for all E Cosder the deg[] = m hece by defto deg + E = m ths mples that E = m for all E herefore E s costat Remark 34 he coverse of above theorem eed ot to be true geeral Eample 35 Cosder a SVHG H = X E where X = {a b c d} ad E = {P Q R S} whch s defed by P = {a 8 3 b 8 3} Q = {b 8 3 d 8 3} R = {c 8 3 d 8 3} S = {d 8 3 a 8 3} Here E s costat but dega = ad degd = herefore dega ad degd are ot equal hece H s ot regular SVHG lso deg[a] = ad deg[d] = 3 8 thus deg[a] ad deg[d] are ot equal hece H s ot totally regular SVHG we coclude that H s ether regular ad or totally regular SVHG heorem 36 Let H = X E be a SVHG the E s costat o X f ad oly f followg are equvalet H s regular SVHG H s totally regular SVHG Proof: Suppose H = X E be a SVHG ad E s costat H the E = c = c c c 3 for all E Suppose H s a -regular SVHG the deg = = 3 for all E Cosder deg[] = deg + E = + c for all E Hece H s totally regular SVHG et suppose that H s a m-totally regular SVHG the deg[] = m = m m m 3 for all E that s deg + E = m for all E hs mples that deg = m c for all E hus H s regular SVHG Coversely: Suppose cotrary E s ot costat that s E ad E y ot equal for some ad y X Let H = X E be a -regular SVHG the deg = = 3 for all E Cosder deg[] = deg + E = + E deg[y] = degy + E y = + E y sce E ad E y are ot equal for some ad y X hece deg[] ad deg[y] are ot equal thus H s ot totally regular SVHG whch cotradct to our assumpto et let H be totally regular SVHG the deg[] = deg[y] that s deg + E = degy + E y deg degy = E y E sce RHS of above equato s ozero hece LHS of above equato s also ozero thus deg ad degy are ot equals so H s ot regular SVHG whch s aga cotradct to our assumpto thus our supposto was Muhammad slam Malk l Hassa Sad roum loret Smaradace Regular Sgle Valued eutrosophc Hypergraphs

22 eutrosophc Sets ad Systems Vol 3 06 wrog hece E must be costat ths completes the proof Defto 37 Let H = X E be a regular SVHG the the order of SVHG H whch s deoted ad defed by OH = p q r where p = E X q = E X r = E X for every X ad the sze of regular SVHG whch s deoted ad defed by SH = = S E where SE = a b c whch s defed by a = b = c = E E E E E E Eample 38 Cosder the SVHG H = X E where X = {a b c d} ad E = {P Q R S} whch s defed by P = {a 8 3 b 8 3} Q = {b 8 3 c 8 3} R = {c 8 3 d 8 3} S = {d 8 3 a 8 3} Here the order ad the sze of H are gve 3 8 ad respectvely Proposto 39 he sze of a -regular SVHG H = X E s k/ where X = k Proposto 30 Let H = X E be a m-totally regular SVHG the SH + OH = mk where X = k Corollary 3 Let H = X E be a -regular ad a m-totally regular SVHG the OH = km - where X = k Proposto 3 he dual of a -regular ad a m-totally regular SVHG H= X E s aga a -regular ad a m- totally regular SVHG Defto 33 he SVHG s sad to be complete SVHG f for every X = {: X-{}} that s cotas all remag vertces of X ecept Eample 34 Cosder a sgle valued eutrosophc hypergraphs H = X E where X = {a b c d} ad E = {P Q R} whch s defed by P = {a c 8 3} Q = {a b 8 d 8 } R = {c d 7 b 4 } Here a = {b c d} b = {a c d} c = {a b d} d = {a b c} Hece H s complete SVHG Remark 35 a complete SVHG H = X E the cardal-ty of s same for every verte heorem 36 Every complete SVHG H = X E s both regular ad totally regular f s costat H Proof: Let H = X E be a complete SVHG suppose E s costat H so E E = c = c c c 3 sce SVHG s complete the by defto for every verte X = { : X-{}} the ope eghbourhood degree of every verte s same Hece deg = = 3 for all E Hece complete SVHG s regular SVHG lso deg[] = deg + E = + c for all E hus H s totally regular SVHG Remark 37 Every complete SVHG s totally regular eve f E s ot costat Defto 38 he SVHG s sad to be k-uform f all the hyperedges have same cardalty Eample 39 Cosder a SVHG H = X E where X = {a b c d} ad E = {P Q R} whch s defed by P = {a 8 3 b 7 5 3} Q = {b 8 8 c 8 4 } R = {c 8 4 d 8 9 5} Muhammad slam Malk l Hassa Sad roum loret Smaradace Regular Sgle Valued eutrosophc Hypergraphs

23 eutrosophc Sets ad Systems Vol Cocluso heoretcal cocepts of graphs ad hypergraphs are hghly utlzed by computer scece applcatos he SVHG are more fleble tha fuzzy hypergraphs ad tutostc fuzzy hypergraphs he cocepts of SVHGs ca be appled varous areas of egeerg ad computer scece ths paper we defed the cocept of regular ad totally regular SVHGs We pla to eted our research work to regular ad totally regular o polar SVHGs regular ad totally regular o terval valued eutrosophc hypergraphs rregular ad totally rregular o SVHGs rregular ad totally rregular o bpolar SVHGs Refereces [0] Pradeepa ad SVmala regular ad totally regular tutostc fuzzy hyper graphs teratoal joural of mathematcs ad applcatos volume 4 ssue -C [] V Devadoss Rajkumar & J P Praveea Study o Mracles through Holy ble usg eutrosophc Cogtve Maps CMS : teratoal Joural of Computer pplcatos [] agoor Ga ad M hamed Order ad Sze uzzy Graphs : ullet of Pure ad ppled Sceces Vol E o [3] Ga ad S Shajtha egum Degree Order ad Sze tutostc uzzy Graphs : tl Joural of lgorthms Computg ad Mathematcs [4] agoor Ga ad SR Latha O rregular uzzy Graphs : ppled Mathematcal Sceces Vol 6 o [5] Smaradache Refed Lteral determacy ad the Multplcato Law of Sub-determaces : eutrosophc Sets ad Systems Vol [6] Smaradache ypes of eutrosophc Graphs ad eutrosophc lgebrac Structures together wth ther pplcatos echology Semar Uverstatea raslvaa d rasov acultatea de Desg de Produs s Medu rasov Romaa 06 Jue 05 [7] Smaradache Symbolc eutrosophc heory russels: Europaova p [8] Smaradache eutrosophc set - a geeralzato of the tutostc fuzzy set : Graular Computg 006 EEE tl Coferece DO: 009/GRC [9] H Wag Smaradache Y Zhag ad R Suderrama Sgle Valued eutrosophc Sets : Multspace ad Multstructure [0] H Wag Smaradache Zhag Y-Q ad R Suderrama terval eutrosophc Sets ad Logc: heory ad pplcatos Computg Phoe: Hes 005 [] urkse terval valued fuzzy sets based o ormal forms : uzzy Sets ad Systems vol [] K taassov tutostc fuzzy sets : uzzy Sets ad Systems vol [3] K taassov ad G Gargov terval valued tutostc fuzzy sets : uzzy Sets ad Systems vol [4] L Zadeh uzzy sets : formato ad Cotrol [5] P hattacharya Some remarks o fuzzy graphs : Patter Recogto Letters [6] R Parvath ad M G Karuambga tutostc uzzy Graphs : Computatoal tellgece : heory ad applcatos teratoal Coferece Germay Sept [7] R orzooe H Rashmalou More Results O Vague Graphs UP Sc ull Seres Vol 78 ssue [8] S roum M alea Smaradache akal Sgle Valued eutrosophc Graphs: Degree Order ad Sze UZZ EEE Coferece 06 8 page [9] S roum M alea akal Smaradache Sgle Valued eutrosophc Graphs : Joural of ew heory o [0] S roum M alea akal Smaradache O polar Sgle Valued eutrosophc Graphs : Joural of ew heory o [] S roum M alea akal Smaradache terval Valued eutrosophc Graphs SSOM Coferece 06 press [] S roum Smaradache M alea akal troducto to polar Sgle Valued eutrosophc Graph heory OPRO coferece 06 [3] S roum M alea akal Smaradache Operatos o terval Valued eutrosophc Graphs 06 submtted [4] S roum M alea akal Smaradache Strog terval Valued eutrosophc Graphs 06 submtted [5] S Mshra ad Pal Product of terval Valued tutostc fuzzy graph : als of Pure ad ppled Mathematcs Vol 5 o [6] S Rahurkar O solated uzzy Graph : tl Joural of Research Egeerg echology ad Maagemet 3 pages [7] W Vasatha Kadasamy K latheral Smaradache eutrosophc Graphs: ew Dmeso to Graph heory Kdle Edto 05 Muhammad slam Malk l Hassa Sad roum loret Smaradace Regular Sgle Valued eutrosophc Hypergraphs

24 eutrosophc Sets ad Systems Vol [8] Smaradache eutrosophy / eutrosophc probablty set ad logc merca Res Press see pages last edto ole; revewed Zetralblatt fuer Mathematk erl Germay: [9] Des Howe Eglad he ree Ole Dctoary of Computg 999 Receved: December 0 06 ccepted: December 8 06 Muhammad slam Malk l Hassa Sad roum loret Smaradace Regular Sgle Valued eutrosophc Hypergraphs

25 eutrosophc Sets ad Systems Vol Uversty of ew Meco ragular Dese uzzy eutrosophc Sets Sujt Kumar De ad smat eg Departmet of Mathematcs Mdapore College utoomous 70 West egal da E-mal: skdemamo008com@gmalcom Cetre for Mathematcs ad Statstcal Sceces Lahore School of Ecoomcs 5300 Lahore Paksta E-mal: beg@lahoreschooledupk bstract ths study we troduce the cocept of deser property fuzzy membershp fucto used eutrosophc sets We preset several ew deftos ad study ther propertes Defuzzfcato methods over eutrosophc tragular dese Keywords: Dese fuzzy set; ragular dese fuzzy eutrosophc set; defuzzfcato method fuzzy sets ad eutrosophc tragular tutostc dese fuzzy sets are the gve ally practcal applcablty of the methods have bee dscussed wth graphcal mplcatos recet tmes troducto or ay kd of mult-attrbute decso makg MDM t s essetal to have adequate crsp data but moder stuatos crsp data are adequate he data for whch they used to rely more are bascally mprecse vague approprate ad pecewse utruth as a whole elap [3] made a attempt to study wth the four valued logc amely ruth false Ukow U ad Cotradcto C He used a b-lattce where the four compoets were ter-related Smaradache [3] fouded ad developed the eutrosophc set eutrosophc logc eutrosophc probablty ad eutrosophc statstcs Several researchers Ye [9] swas et al [45] Madal ad Pramak [0] etc have dscussed several rakg method based o curret problems usg eutrosophc sets S lso recetly the multvalued power operator S has bee veveloped by Peg at al[]uzzy set theory was frst studed by Zadeh [] but after few decades later the cocept o hestat fuzzy set has bee grow by orra[5] Moreover tutostc fuzzy evromet umerous research artcles have bee studed by emet practtoer he cocept o tutostc fuzzy sets S has bee developed depedetly by taassov[] ad Dubos et al [9] hrough ts processwag et al [78] Pe ad Zheg [] dscussed ew cocepts o evdece based S ad a ovel approach for decso makg respectvely However the decso maker s DM are usually applyg ther approprate membershp grade values of the dfferet attrbutes whch are pror ad epereced data ut realty the data predcted a day ago may ot be useful for tomorrow ad may cases those grade values demadg chagg values wth the chage of the dealg frequecy amog several moopoly eterprses or betwee the tme gap also hus t s troublesome to fd the actual data because most of the orgal data s hdde ad secret uder some atoal or teratoal law ad orders or stace to fd the formato over flood vctms a partcular place several opos may come out ut the data accepted by the authortes usually vary the realty because of the lmtatos o govermetal facal supports to be offered to the vctms However the stuato bega to clear as the day passg o o model the above stuato ths artcle we frst gve some basc cocepts o eutrosophc set S ad the we develop the S uder dese fuzzy evromet he fuzzy compoets uder several compostos are dscussed from the estg lterature et some etesos are made wth proper justfcato he paper s orgazed as follows: Secto descrbes some basc cocepts of S for subsequet use secto 3 we develop the S uder dese fuzzy evromet Secto 4 deals wth defuzzfed values of S Secto 5 mproves S assessmet uder dese fuzzy evromet Secto 6 gves further mplcatos of dese fuzzy S Secto 7 presets applcatos to show the practcalty ad feasblty of our method Secto 8 eds the paper wth some cocludg remarks Prelmares Here we shall dscuss some basc cocepts ad operatos o eutrosophc set Defto [4] Let X be a space of pots objects wth geerc elemet he a eutrosophc set S X s characterze by a truth membershp fucto a determacy membershp fucto ad a falsty membershp fucto t s deoted by s =< > where the fuctos ad are real stadard or o-stadard subsets of ]0 + [ hat s X ]0 + [ X ]0 + [ ad s X ]0 + [ satsfyg the relato s 0 sup + sup + sup 3 + S K De ad eg ragular dese fuzzy eutrosophc Sets

26 eutrosophc Sets ad Systems Vol Defto [3]he complemet c of a S s defed as follows: c = { + } c = { + } ; c = { + } Defto 3 [3] eutrosophc set s cotaed other eutrosophc set e f ad oly f the followg results hold good for X f f sup sup f f sup sup f f sup sup c Defto 4 [6]he complemet s of a sgle valued S s gve by s c = s c s c = s ; s c = s Defto 5 [6]he uo of two sgle valued eutrosophc sets ad deoted by C = ts truth membershp determacy membershp ad falsty membershp fuctos are related to those of ad as follows: C = ma C = ma C = m X Defto 6 he tersecto of two sgle valued eutrosophc sets ad deoted by C = ts truth membershp determacy membershp ad falsty membershp fuctos are related to those of ad as follows: C = m C = m C = ma X Defto 7 he addtoof two sgle valued eutrosophc sets ad deoted by C = ts truth membershp determacy membershp ad falsty membershp fuctos are related to those of ad as follows: C = + C = C = X Defto 8 he multplcato of two sgle valued eutrosophc sets ad deoted by C = ts truth membershp determacy membershp ad falsty membershp fuctos are related to those of ad as follows: C = C = + C = + X Remark eutrosophc Cube descrbg S &S Jea Dezert [8] troduced the eutrosophc cube C D E G H to make a dstcto betwee S ad S or techcal use we take the classcal terval [0] for the S parameters ad he the cube CDEGH s called techcal / relatve eutrosophc cube ad ts eteso C D E G H s called the absolute eutrosophc cube ow we dvde the techcal eutrosophc cube to three dsjot regos he observatos from the followg cube are he equlateral tragle DE whose sdes are equal to t represets the geometrcal locus of the pots whose sum of the coordates s hs tragle s kow as taassov-tutostc fuzzy set -S Here f q s a pot o DE or sde of t the as -S t q + q + f q = he pyramd ED[stuated the rght sde of the ED cludg ts faces D base E ad ED lateral faces but ecludg ts face DE] s the locus of the pots whose sum of coordates s less tha f p s pot o ED the t q + q + f q < as S wth complete formato the left sde of DE the cube there s the sold EGCDEDecludg DE whch s the locus of pots whose sum of ther coordates s greater tha as the paracosstet set f a pot r les o EGCDED the t q + q + f q > hus we have a source whch s capable to fd oly the degree of membershp of a elemet; but t s uable to fd the degree of o-membershp other source s capable to fd oly the degree of o-membershp of a elemet Or a source whch oly computes the determacy Puttg these results we always havet q + q + f q Moreover formato fuso whe dealg wth determate models that s elemets of the fuso space whch are determate/ukow such as tersectos we do t kow f they are empty or ot sce we do t have eough formato smlarly for complemets of determate elemets etc ; f we compute the beleve that elemet truth the dsbeleve that elemetfalsehood ad the determacy part of that elemet the the sum of these three compoets s strctly less tha the dfferece to s the mssg formato hs s show g S K De ad eg ragular dese fuzzy eutrosophc Sets

27 eutrosophc Sets ad Systems Vol ragular dese fuzzy evromet Here we dscuss the dese fuzzy evromet the all possble cases of S Defto 9 [7]Let be the fuzzy umber whose compoets are the elemets of R beg the set of real umbers ad beg the set of atural umbers wth the membershp grade satsfyg the fuctoal relatoμ R [0] ow as f μ for some Rthe we call the set as dese fuzzy set f s tragular the t s called DS ow f for some μ attas the hghest membershp degree the the set tself s called ormalzed ragular Dese uzzy Set or DS he graphcal terpretato s show g G X C G C μ a a a H D0 0 D E E g-: Geometrc represetato of eutrosophccube g: Membershp fucto of DS based o defto 9 Eample s per deftos 9 let us assume the DS as follows =< a ρ + a a + σ + > for 0 < ρ σ < he membershps fucto for 0 s defed as follows:γ = lows:γ = H a m { 0 f < a ρ + ad > a + σ + { a ρ + ρa + { a + σ + σa + } f a ρ + a } f a a + σ + Smlarly here also we ote that the ordary membershp fuctos of falsehood ad determacy s gve by γ = { 0 f < b ρ + ad > b + σ + γ = { { b } f b ρb ρ b + + { b } f b σb b + σ f < c ρ 3 + ad > c + σ 3 + { c ρ3c } f c ρ 3 c + + { c σ3c } f c c + σ Defto 0: DS based o o-membershp & determate fucto Let be the fuzzy umber whose compoets are the elemets of R whose o-membershp grade satsfyg the fuctoal relato θ R [0] ow as f θ 0 for some Rthe we call the set as dese fuzzy set f we cosder the fuzzy umber of the form = a a a 3 the we call t ragular Dese uzzy Set ow f for = 0 θ attas the hghest membershp degree the we ca epress ths fuzzy umber as ormalzed ragular Dese uzzy Set or DS Eample-: Let the falsty set s gve by =< b ρ e b e b + σ e > for 0 < ρ σ < 5 d ts o-membershp fucto for 0 s defed as θ = 3 4 S K De ad eg ragular dese fuzzy eutrosophc Sets

28 eutrosophc Sets ad Systems Vol { 0 f < b ρ e ad > b + σ e { b e ρ b e } f b ρ e b e { b e σ b e } f b e b + σ e 6 he graphcal represetato of o-membershp fucto s gve g 3 Defto-: eutrosophc set X s sad to be eutrosophc tutostc Dese fuzzy Set f the elemets of S that s the fuctoal compoets ad are take from the real stadard subsets of [0 ] hat s X [0 ] X [0 ] ad s X [0 ] havg the property that as f 0 satsfyg the relato s 0 sup + sup + sup 3 Remark he S for depedecy compoetsye [0] X θ a a a 3 0 a m Here we draw the smple Ve dagram for the S wth depedecy compoets We use ths dagram to realze the overall assessmet of the fuzzy compoets ccordg to probablty theory the overall score obtaed from the fg-4 s stated 4 ote that f a = b = c holds the fuzzy sets stated 5 ad 7 the the commo rego of that set wll be crsp oe hs s show g-4 g 3: o membershp fucto of DS Eample-3: Let the determacy dese fuzzy set be of the form C =< c ρ 3 e c e c + σ 3 e > for 0 < ρ 3 σ 3 < 7 Wth the membershp fucto { π = 0 f < c ρ 3 e ad > c + σ 3 e { c e ρ 3 c e } f c ρ 3 e c e 8 { c e } f σ 3 c e c e c + σ 3 e Defto-: Let X be a space of pots objects wth geerc elemet he a eutrosophc set X s characterze by a truth membershp fucto a determacy membershp fucto ad a falsty membershp fucto he fuctos ad are real stadard or o-stadard subsets of ]0 + [hat s X ]0 + [ X ]0 + [ ad s X ]0 + [ havg the property that as f 0 d satsfyg the relato s 0 sup + sup + sup 3 + Crsp 4 Some bascs over epected values of S Here we shall dscuss over the ultmate score or epected defuzzfed values for the proposed eutrosophc set s =< > Whe the compoets ad are depedet he as per [swas et al[4]] the total average epected score wll be S = 3 { + + }9 d the truth favorte relatve epected value score s gve by g-4: Ve dagram of Geeral eutrosophc Set S K De ad eg ragular dese fuzzy eutrosophc Sets

29 eutrosophc Sets ad Systems Vol S = or formato lackg S ths part ca be dvded to several sub cases a he compoets ad all together costtute a postve skewed dstrbuto ths case truth leads the major part the total score wll be S = w + w + w 3 for w > w > w 3 ad w + w + w 3 = b he compoets ad all together costtute a egatve skewed dstrbuto ths case falsehood leads major part the total score wll be S = w + w + w 3 for w < w < w 3 ad w + w + w 3 = c he compoets ad all together costtute a ormal dstrbuto ths case truth ad falsehood are symmetrc the total score wll be S = w + w + w 3 for w < w > w 3 ad w + w + w 3 = 3 Whe the compoets ad are depedet a f the compoets keep the postve sg the the ultmate score s gve by S = b or the case of S the effect of {0} or ukow ad the sg of be egatve ad hece the ultmate score be S = + 5 c [Che ad a [6]] Sce the epected values of < ad < so ther product values ad hece ther effect ca be gored ths case the score fucto be S = 6 5 mproved S assessmets uder dese fuzzy evromet Case- : Whe all the membershp fuctos keep ther values of smlar types for same umber of observatos/teractos or tme duratos rst of all we shall dscuss the fuzzy assessmets uder dese fuzzy developed by De ad eg [7] he basc am of the dese fuzzy model s that each ad every fuzzy compoet reaches to sgleto crsp value wheever we would lke to eperecg wth fuzzy data for a log perod of tme or teractos practce Here also we assume the learg epereces are coducted wth the same elapsed tme or teractos for all the fuzzy compoets of S hus ulke swas et al [4]; utlzg dese cocept ad α cuts developed by De ad eg[7] the defuzzfed value reduces from the formula = α=t= {Lα t + α=0t=0 Rα t} dαdt = a { + σ ρ Log + } for tme 4 depedet fuzzy membershp fucto ad that for frequecy depedet membershp fucto : = a { + σ ρ } = a =0 [ + + σ ρ { }]ad obtaed as or tme depedet reduces to S t = {[ 3 t + [ t] + [ t]} S = [a 3 { + σ ρ Log + } + 4 b { + σ ρ Log + } + 4 c { + σ 3 ρ 3 Log + }] 7 4 d that for frequecy depedet 3S = [ + [ ] + [ ] S = a 6 { + σ ρ } + =0 b + 6 =0 { + σ ρ } + c + 6 { + σ 3 ρ 3 =0 } or tme depedet fuzzy compoets 0 reduces to S = 3a { + σ ρ Log + }/ 4 [a { + σ ρ Log + } + 4 b { + σ ρ Log + } + 4 c { + σ 3 ρ 3 Log + }]9 4 d that for frequecy depedet S = 3 a { + σ ρ + } / =0 [ a { + σ ρ } + =0 b + =0 { + σ ρ } + c + { + σ 3 ρ 3 =0 } ] 0 + Smlarly for -3 we get 53 or tme depedet -3 reduces to S t = w [ t + w [ t] + w 3 [ t] S = w a { + σ ρ Log + } + 4 w b { + σ ρ Log + } + w 3 c { + σ 3 ρ Log + }wthw + w + w 3 = S K De ad eg ragular dese fuzzy eutrosophc Sets

30 eutrosophc Sets ad Systems Vol d that for frequecy depedet S = w a { + σ ρ =0 } + + w b { + σ ρ =0 } + + w c 3 { + σ 3 ρ 3 =0 } wth w + + w + w 3 = 54 or frequecy depedet fuzzy compoets4 reduces to S = S a b c = a { + σ ρ + } =0 + b { + σ ρ + } =0 + c { + σ 3 ρ 3 + } =0 a b [ + σ +σ ρ ρ + =0 + ρ ρ + σ σ 6 + ] b c [ + σ +σ 3 ρ ρ 3 + =0 + ρ ρ 3 + σ σ ] a c [ + σ +σ 3 ρ ρ 3 + =0 + ρ ρ 3 + σ σ ] + a b c [ + σ ρ + + ρ + σ +σ 3 σ ρ ρ 3 ] + =0 + a b c =0 [ ρ ρ +ρ 3 + σ σ +σ σ σ σ 3 ρ ρ ρ ] ρ ρ +ρ ρ 3 ρ ρ 3 +σ σ +σ σ 3 σ σ σ σ σ 3 +ρ ρ ρ 3 ] 4+ 3 [ =0 3 for detal see apped equato - 5 lso the results for tme cotuous fuzzy membershp we have by smply tegratg the above sum wth respect to tme ad applyg the dese rule ad get S = a { + σ ρ Log + } 4 + b { + σ ρ Log + } 4 + c { + σ 3 ρ 3 Log + } 4 a b [ + σ +σ ρ ρ Log + ρ ρ + σ σ 4 + ] b c [ + σ +σ 3 ρ ρ 3 Log + 4 ρ ρ 3 + σ σ ] a c [ + σ +σ 3 ρ ρ 3 Log ρ ρ 3 + σ σ 3 + ] +a b c [ + σ ρ Log + + ρ + σ +σ 3 σ ρ ρ 3 Log 4 + ρ ρ +ρ 3 + σ σ +σ 3 ] 4 + +a b c [ ρ ρ +ρ ρ 3 ρ ρ 3 +σ σ +σ σ 3 σ σ σ σ σ 3 ρ ρ ρ σ σ σ 3 +ρ ρ ρ 3 6+ ] 4 55 or tme depedet fuzzy compoets 5 reduces to S = + S = a { + σ ρ Log + } 4 + b { + σ ρ Log + } 4 a b [ + σ +σ ρ ρ Log + ρ ρ +σ σ ] d that for frequecy depedet fuzzy compoets S = a { + σ ρ + } =0 + b { + σ ρ + } =0 a b [ + σ +σ ρ ρ = ρ ρ +σ σ 6 + ] S K De ad eg ragular dese fuzzy eutrosophc Sets

31 eutrosophc Sets ad Systems Vol Case- : Whe all the membershp fuctos keep ther values of dfferet types for same umber of observatos/teractos or tme duratos Here we shall take the membershp fuctos of ad as stated 6 ad 8 { S t ow the score fucto of 6 uder tme sestve fuzzy umbers s gve by S t = a ρ +t ρa +t a + σ +t σa +t b e t b ρ e t f u v say b e t b σ e t fv w say [ + t + et ] + t + f u v say ρ a ρ b ρ ρ = + t + + [ + t + et ] fv w say { σ σ σ a σ b Usg α cuts ad employg the de formula developed by De ad eg [7] we get S = α=t= {Lα t + Rα t} dαdt α=0t=0 +t = [ ρ + ρ +t σ + + σ t et +t et +t et +t σa + et σb ρa + ρb ] dt = a b σa + σb ρa + ρb [+tρ +ρ ρρ + 0 +tρ b +e t ρ a +tσ +σ + σ σ +tσ b +e t σ a ] dt 7 d that for dscrete case replacg t by we wrte the epected score of the S as S = a b [+ρ +ρ ρρ =0 + +ρ b +e ρ a +σ +σ + σ σ +σ b +e σ a ] 8 Case-: f we assume that the learg effects are ot performed same tme durato / teractos for all fuzzy compoets the the above defuzzfcato formula 7 & 8 reduces to 3 S = a { + σ ρ 4 Log + } + b { + σ ρ 4 Log + } + c { + σ 3 ρ Log + 3 }9 d that for frequecy depedet 3S m p = [ m + [ ] + [ p] S = M a 6M { + σ ρ } + m=0 b +m 6 =0 { + σ ρ } + P c + 6P { + σ 3 ρ 3 p=0 } 30 +p he detaled dscusso s made pped 6 mplcato of Dese fuzzy S S the tradtoal cocept of membershp values of truth falsehood ad determacy are ether fed or caot be chaged oce t s assged ut the preset study reveals that such pots are cotuously chagg because of learg epereces or fleblty of the huma behavor ad tetos or stace whe we ask a questo to a perso about the lfe loss due to a certa tra accdet the s he mght be aswered that 75% of the whole people ded 40% ot ded ad 30% s ukow ut after few days later f the same questo has bee throw to the same perso the obvously hs/her aswer mght dffer from the earler statemet hs may be 90% 50% ad 0% or 0% 80% 5% Such kd of observato occurs due to formato gatherg from the socety or learg epereces as soo as the tme s passed/ creased frequecy of huma teractos wth the localty/ socety/ mass meda etc y ths way before gog to take govermetal supports Decso maker s accoutablty o the bass of that pror formato several equres commttee wll be formed ad fally come to a cocrete decso for establshg law ad order people s beeft he earler cocepts o S aalyze the data based o frst aswer obtaed from that perso but our preset study t aalyzes the data subsequetly obtaed 6 Procedure for the computato of defuzzfed values of a eutrosophc SetsSs Step-: d the S volved the dfferet feld of actvtes Step-: d the approprate membershp compoets of that Ss Step-3: d the teror relatoshp amog the dfferet Ss Step-4: Select the strateges volved those Ss a f we are tedg to fd the jot performaces the take ther uo usg the defto 5 S K De ad eg ragular dese fuzzy eutrosophc Sets

32 eutrosophc Sets ad Systems Vol b f we are tedg to fd the commo performaces the take ther tersecto usg the defto 6 c f the selecto of oe s S mght able to chage the other s S the to fd the completes volved take ther multplcato usg the defto 8 d f the selecto of oe s S do ot able to chage the other s S at all the to fd the completes volved take ther addto usg the defto 7 Step-5: fter gettg the approprate S obtaed from Step-4 use defuzzfcato rules whch oe you are gog to assess 7 pplcatos of dese fuzzy S DS Several applcatos ca be draw from our day to day lfe problems from scece ad egeerg socology phlosophy crme research educatoal psychology etc he followg are some major areas where the DS ca be appled 7 ay kd of decso makg process Eample: Suppose a supply cha SC model the set of formato ss =< s s s > ad sr =< r r r > for both the suppler ad retaler are avalable uder dese fuzzy evromet rst of all we have to obta the bouds of each fuzzy compoets utlzg dese property; the check whether these sets are subsets of each other or ot ow applyg cogruecy rule or smlarty measures the score fuctos for the cha ca be obtaed ad ca be solved by the proposed defuzzfcato methods ote that f these two sets are dsjot the the cha mmedately gets breakg dow ad the decso maker wll have to choose aother model as well 7 Psychologcal testg/ mltary selecto Eample: Suppose a psychologcal test there are fve dfferet attrbutes to be measured he attrbutes are {Moral value behavor leadershp crmal offece resposblty} mog these attrbutes some of them have postvely correlato some of them are egatvely correlato ad rest of them has o relato However we have to perform the membershp fuctos of each attrbutes uder dese fuzzy evromet ad eercsg these every after some stpulated terval of tme/ days ake for stace to ga best leadershp qualty oe mght have to compromse wth crmal actvtes ad bad resposblty may cases practce O the other had a perso havg good moral stadards she mght carry a good behavor ad good leadershp Uder these crcumstaces wheever we wsh to compute the total score few of the score compoets of the S became egatve hus to get the overall performace the proposed defuzzfcato method ca be appled ad raked accordgly 73 Leadershp assessmet uder socal ageda Eample: Suppose a ope problem o flood proe zoe a localty has bee throw before two poltcal leaders havg dfferet deologes he problem tself cotas three dfferet parametrc attrbutes lke {hghly flood proe ukow o flood at all} ths case frst set s a approprate S for the gve attrbutes uder dese fuzzy evromet he we ask to aswer the questo to the poltcal leaders at dfferet places ad dfferet terval of tmes We usually otce that the scores obtaed by them at dfferet tme are ot the same hus to have the actual score obtaed by each of them we mght have to rely ther vews that were delvered at the fal stage of assessmet 74 ssessg the age of a dgtal mage Let a dgtal mage be three parametrc attrbutes to be measured hese attrbutes are {brghtess whte darkess} rst of all wth proper deftos costruct the membershp fuctos of each of these attrbutes uder dese fuzzy evromet ow chagg tme wth proper record compute score values of the S ad compare wth the values obtaed from the specme dgtal mage each tme Cotue ths process utl the epected value gets merged wth the values of the specme mage ally get the age the tme you have recorded last tme 75 Vulerablty/ rsk assessmet dsaster proe zoe Let the attrbutes uder study for measurg the vulerablty or rsk a dsaster maagemet s {whether zoe s dsaster proe surace of lfe S K De ad eg ragular dese fuzzy eutrosophc Sets

33 eutrosophc Sets ad Systems Vol covered valuato of the property} May tmes several atttudes may come whether a partcular zoe s dsaster proe or ot he lfe volved that area s kow ad hece the surace covered s assessed but valuato of the property s qute uclear practce However f we thk of brds/ amals lke pet ad frm amals lfe surace or beyod the ths part also carry some formato lack Uder these crcumstaces computg score values of the proposed S at several years the actual rsk ca be measured he estg research may be vewed akacs [4] 76 Ecoomc Cultural Poltcal Clmatc Dsease Mappg Wth the help of S we ca map wth Coutry/State eve the world also wth respect to dfferet soco ecoomc parameters or eample studyg wth {rch medocre poor} dfferet coutres of the world the ecoomcally soud/usoud such as developed uder developed less developed coutres ca be detfed globally ad ca map them accordgly Smlarly for peoples cultural etty { true tellg o tellg le tellg} or { lve to eat o commets eat to lve} or { lke dace lke sog both oe } ca be measured at dfferet tme at dfferet coutres ad the map accordgly or poltcal allace takg 0 years data from dfferet cou-try peoples atttudes o {democracy autocracy dealsm materalsm ad socalsm} may be cosdered for better mappg of fredshp or mappg o dfferet clmatc zoe the parameters lke { hot ether hot or cold cold} or { hghly polluted ukow o polluto} ad that for mappg of severe dsease proe zoe takg 5-0 years data of publc opo o { yphod Darrhea cholera free of dsease } etc ca be appled developg S 77 Game theory ths compettve world behd ay kd of actvtes there must have a hdde game or each strategc player we may thk of a S havg three or more fuzzy compoets hese compoets are chagg for several reasos or stace sudde fall of share market stat prce hke of commodtes ammedmet of Govt polces etc may cause the fleblty of fuzzy o membershp grade However the players mght S K De ad eg ragular dese fuzzy eutrosophc Sets have to chage ther pla wth oe day durato Utlzg S fuzzy cross product ad algebrac propertes the problem ca be solved o kow the ultmate ga of each player our proposed defuzzfcato method ca be appled 78 Persoalty est/ ablty detfcato or the detfcato of psychologcal ablty or cogtve developmet several possble attrbutes are take for a dvdual he costructg dfferet membershp epected eutrosophc sets are draw Defuzzfyg the gve S we may easly measure the dfferet abltes whch may gude to form a persoalty developmet Cocluso ths study we have eplaed the estg eutrosophc set uder dese fuzzy evromet radtoally most of the researchers were eperecg wth the o membershp grade value drectly from the study area ad took decsos through some aggregato rules or rakg scores hey do ot feel the urge to defuzzfy the S earler lso some of the cases fuzzy graph theory or fuzzy matrces have bee developed to capture the decso theory he cocepts of huma learg the chagg characterstcs of the fuzzy membershp wth respect to tme ad the umber of observatos have bee gored by the fouder thkers of S We eed to defuzzfy the S uder dese fuzzy evromet for the followg reaso: a he fuzzy elemets S are obtaed drectly from feld data oly b he fuzzy fleblty may chage wth tme elapsed ad teractos covered c o kow the dvdual value rather tha membershp degree because a dfferet value may carry the same degree but the reverse s ot true due to covety property d o realze the actual world rather tha a hypothetcal world pped o fd the values of the followg equato 5 S = for the cases of geeral membershp fuctos We take the help of α cuts of the fuzzy compoets ow as per De ad eg[7] the left ad rght α cuts of the fuzzy set = a a a 3 or =<

34 eutrosophc Sets ad Systems Vol a ρ a + a + σ > for 0 < ρ + σ < s gve by L α = a ρ + ρ α ad + + R α = a + σ + σ α + ow from the propertes ofα cuts we have α = α α = {a ρ + + ρ α + a + σ + σ α + } {b ρ α + b + σ α + } = a b { ρ + + ρ α + ρ α + + σ + σ α + + σ α + } = a b [ ρ + + α { ρ ρ + + ρ ρ + } ρ ρ α + + σ + + α { σ σ + + σ σ + } σ σ α + ] ote that above we assume the left ad rght α cuts are creasg ad decreasg fuctos ofα respectvely f t s mpossble to determe whether the above codtos hold or ot the the gross value of the α cuts are gve by he left α cut of α = = M {a ρ + + ρ α + a + σ + σ α + } {b ρ α + b + σ α + } = Ma b { ρ + + ρ α + ρ α + ρ + + ρ α + + σ α + + σ + σ α + ρ α + + σ + σ α + + σ α + } d the rght α cut of α = = Maa b { ρ + + ρ α + ρ α + ρ + + ρ α + + σ α + + σ + σ α + ρ α + + σ + σ α + + σ α + } he same rule could be appled other cases also herefore the de value s gve by = = a b [ ρ + =0 + { ρ ρ + + ρ ρ + } ρ ρ σ + + { σ σ + + σ σ + } σ σ 3 + ] a b [ + σ +σ ρ ρ =0 Smlarly + + ρ ρ +σ σ 6 + ] = a c [ + σ +σ 3 ρ ρ 3 =0 + + ρ ρ 3 +σ σ ] lso to fd we wrte α = α α = {b ρ α + b + σ α + } {c ρ 3α + c + σ 3α + } = b c [ ρ α + ρ 3α + + σ α + + σ 3α + ] = b c [ α ρ +ρ 3 + ρ ρ 3 α + α + + σ +σ σ σ 3 α + ] herefore = b c [ + σ +σ 3 ρ ρ 3 + ρ ρ 3 +σ σ 3 =0 ] Moreover to fd the de value of we wrte the α = α α α S K De ad eg ragular dese fuzzy eutrosophc Sets

35 eutrosophc Sets ad Systems Vol = a b c [ ρ + + ρ α + + σ + σ α + ] [ α ρ + ρ ρ ρ 3 α + + α σ + σ σ σ 3 α + ] = a b c [ ρ + α + {ρ ρ ρ ρ ρ +ρ 3 } + α { ρ ρ ρ ρ ρ +ρ ρ 3 ρ ρ 3 + } + ρ ρ ρ 3 α σ + + α {σ +σ 3 σ + hus = + σ σ +σ 3 + } +α { σ σ σ σ σ 3 σ σ σ σ 3 + } σ σ σ 3 α ] a b c [ + σ ρ =0 ρ ρ +ρ 3 +σ σ +σ ρ +σ +σ 3 σ ρ ρ ρ ρ +ρ ρ 3 ρ ρ 3 +σ σ +σ σ 3 σ σ 3 + σ σ σ 3 ρ ρ ρ σ σ σ 3 +ρ ρ ρ ] 4 pped Here we shall study the α cuts of ad whe all the compoets occur proper dese property of fuzzy sets[ stated 6 ad 8] ths case the α cut of that s α wll remas the same ut theα cuts of α are gve by { b e ρ b e } α b e αρ ad { b e } α σ b e b e + ασ hus α = [b e αρ b e + ασ ] = b e [ αρ + ασ ] Smlarly α = c e [ αρ 3 + ασ 3 ] + herefore = a { + σ ρ =0 3 = c { + σ 3 ρ 3 } 4 = b { + σ ρ } 5 =0 e =0 e + } ga α = α α = a [ ρ + + ρ α + σ σ α ] {b e [ αρ + ασ ]} = a b e [ ρ + + ρ α + αρ = a b e [ + σ + σ α + + ασ ] ρ + + ρ α + αρ + αρ ρ + α ρ ρ + + σ + σ α + + ασ + ασ σ + α σ σ + ] = a b e [ ρ + α + ρ +ρ ρ ρ + α ρ ρ + ] + σ + α + σ σ σ + σ + α σ σ + 6 hus = a b e [ + σ ρ =0 ρ σ +σ σ +ρ ρ + Smlarly = a c e [ + σ ρ = σ ρ ρ ρ +σ σ ] ρ σ +σ σ 3 +ρ ρ σ 3 ρ 3 ρ ρ 3 +σ σ 3 ]8 3+ or We wrte α = α α = b c e [ αρ + ασ ][ αρ 3 + ασ 3 ] = b c e [ αρ αρ 3 + α ρ ρ 3 + ασ + ασ 3 + α σ σ 3 ] herefore = b c e [ + σ + σ 3 ρ ρ 3 + ρ ρ 3 +σ σ 3 =0 ] 3 = b c [ + σ + σ 3 ρ ρ 3 + ρ ρ 3 +σ σ 3 ] e 3 9 or we take =0 S K De ad eg ragular dese fuzzy eutrosophc Sets

36 eutrosophc Sets ad Systems Vol α = α α α = a b c e [ αρ αρ 3 + α ρ ρ 3 + ασ + ασ 3 + α σ σ 3 ] [ ρ + + ρ α + + σ + σ α + ] = a b c e [ ρ + αρ αρ 3 + α ρ ρ 3 herefore + ρ + α α ρ α ρ 3 + α 3 ρ ρ 3 + σ + + ασ + ασ 3 + α σ σ 3 σ + α + α σ + α σ 3 + α 3 σ σ 3 ] = a b c e [ ρ =0 + ρ ρ 3 + ρ 3 ρ 3 + ρ + ρ 3 ρ ρ 4 ρ σ + σ + + σ 3 + σ 3 σ 3 σ + + σ 3 + σ σ 4 σ 3 ] 0 Hece usg 3-0 the epected values of the gve S ca be obtaed as S = where S = a { + σ ρ + } =0 + b { + σ ρ } e =0 + c { + σ 3 ρ 3 } e =0 a b e [ + σ ρ + =0 + ρ σ + σ σ + ρ ρ + ρ ρ ρ +σ σ 3 + ] b c e [ =0 + σ + σ 3 ρ ρ 3 + ρ ρ 3 +σ σ 3 ] 3 a c e [ + σ ρ + =0 + ρ σ + σ σ 3 + ρ ρ 3 + ρ 3 ρ ρ 3 +σ σ ] + a b c e [ ρ ρ + =0 ρ 3 + ρ 3 ρ 3 + ρ + ρ 3 ρ ρ 4 ρ σ + σ + + σ 3 + σ 3 σ 3 σ σ + σ 3 σ 3 + σ σ 4 σ 3 ] ote that f the fuzzy compoets are epereced wth dfferet teractos the we shall calculate the epected score values as follows: M S = M a { + σ ρ + m } m=0 + b { + σ ρ } e =0 P + c P { + σ 3 ρ 3 } e p p=0 S K De ad eg ragular dese fuzzy eutrosophc Sets

37 eutrosophc Sets ad Systems Vol a b e [ + σ ρ + =0 4 + ρ σ + σ σ + ρ ρ + ρ ρ ρ +σ σ 3 + ] b c e [ =0 + σ + σ 3 ρ ρ 3 + ρ ρ 3 +σ σ 3 ] 3 3 a c e [ + σ ρ 3 + =0 + ρ σ + σ σ 3 + ρ ρ 3 + ρ 3 ρ ρ 3 +σ σ ] + a b c e [ ρ ρ 4 + =0 ρ 3 + ρ 3 ρ 3 + ρ + ρ 3 ρ ρ 4 ρ σ + σ + + σ 3 + σ 3 σ 3 σ σ + σ 3 3 σ + 3 σ σ σ 3 ] Where = mm = m P 3 = mm P ad 4 = mm P Refereces [] Ktaassov tutostc uzzy sets uzzy Sets ad Systems [] Ktaassov tutostc uzzy sets Physca-Verlag Hedelberg Y 999 [3] elap useful four valued logc moder uses of multple valued logcsd Redel ed [4] Pswas S Pramak ad C Gr Cose smlarty measure based mult-attrbute decso makg wth trapezodal fuzzy eutrosophc umbers eutrosophc Sets ad Systems 804a [5] Pswas S Pramak ad C Gr ew methodology for eutrosophc multattrbute decso makg wth ukow weght formato eutrosophc Sets ad Systems 304b 4-5 [6] SM Che ad JM a Hadlg Mult Crtera uzzy Decso Makg Problems ased o Vague Set heory uzzy Sets ad Systems [7] S K De ad eg ragular dese fuzzy sets ad ew defuzzfcato methods Joural of tellget & uzzy systems Press [8] J Dezert Ope questo to eutrosophc fereces t J of Multple-valued logc [9] D Dubos SGottwald Hajek J Kacprzyk ad H Prad ermologcal dffcultes fuzzy set theory he case of tutostc fuzzy sets uzzy sets ad systems [0] KMadal S ad Pramak eutrosophc decso makg model of school choce eutrosophc Sets ad Systems [] ZPe ad L Zheg ovel approach to mult-attrbute decso makg based o tutostc fuzzy sets Epert Systems pplcatos [] J-J Peg J-q Wag X-h Wu JWag ad X-h Che Multvalued eutrosophc Sets ad Power ggregato Operators wth ther pplcatos Mult-crtera Group decso makg problems t J of Computatoal tellgece Systems [3] Smaradache ufyg feld logcs: eutrosophy: eutrosophc probablty set ad logc merca Research Press Rehoboth 998 [4] Makacs Multlevel fuzzy approach to the rsk ad dsaster maagemet cta- PolytechcaHugarca [5] Vorra Hestat fuzzy sets teratoal Joural of tellget Systems; S K De ad eg ragular dese fuzzy eutrosophc Sets

38 eutrosophc Sets ad Systems Vol [6] HWag Smaradache Y Q Zhag R Suderrama Sgle valued eutrosophc set Multspace ad Multstructure [7] J Q Wag R R e H YZhag ad X H Che ew operators o tragular tutostc fuzzy umbers ad ther applcatos system fault aalyss formato Scece [8] J Q Wag P Zhou K J L H Y Zhag ad X H Che Mult crtera de-cso makg method based o ormal -tutostc fuzzy duced geeralzed ag-gregatooperator OP [9] J Ye Mult-crtera decso makg method based o cose smlarty measure betwee trapezodal fuzzy umbers t J of Egeerg scece ad techology [0] J Ye Mult-crtera decso makg method usg the correlato coeffcet uder sgle valued eutrosophc evromet t J of geeral systems [] L Zadeh uzzy sets formato ad Cotrol Receved: ovember 06 ccepted: December 0 06 S K De ad eg ragular dese fuzzy eutrosophc Sets

39 eutrosophc Sets ad Systems Vol Uversty of ew Meco pplcatos of uzzy ad eutrosophc Logc Solvg Mult-crtera Decso Makg Problems bdel asser H Zaed ad Hagar M agub aculty of Computers ad formatcs Zagazg Uversty Zagazg Egypt E-mal: asserhr@zuedueg asserhr@gmalcom aculty of Computers ad formatcs Zagazg Uversty Zagazg Egypt E-mal: hagarmm@gmalcom bstract daly lfe decso makers aroud the world are seekg for the approprate decsos whle facg may challeges due to coflctg crtera ad the presece of may alteratves the way of pursut a powerful decso makg process may researches act mult-crtera decso makg MCDM feld ad may Keywords: uzzy Set heory eutrosophc logc Mult-crtera Decso Makg troducto he mult-crtera decso makg MCDM ca be defed as the process of rakg a set of alteratves ad selectg the most sutable oe based o decso crtera [] Durg the secod half of the 0th cetury MCDM research area has udergoe remarkable ad fast developmet ad may MCDM methods have bee developed to troduce better soluto for mult-crtera decso makg problems [] MCDM process compoets are a set of decso crtera at least two decso makers ad a set of alteratves whch sorted ad raked based o the decso crtera [] Wth a goal of helpg decso makers to rak dfferet alteratves ad choose the best oe that satsfes orgazato s eeds MCDM has bee used to support a wde rage of decsos may areas such as: portfolo optmzato beeft-rsk assessmet techology assessmet ad software selecto [3 4] hs paper aalyses two mult-crtera decso makg methods ad determes ther applcablty to dfferet stuatos by evaluatg ther relatve advatages ad dsadvatages comprehesve lterature revew s coducted to allow a summary of the two methods revew of the use of these methods ad a eamato of the evoluto of ther use s the performed hs paper s orgazed as follows: Secto troduces a bref backgroud of fuzzy set theory uzzy applcatos dfferet MCDM felds are dscussed Secto 3 Secto 4 troduces a bref backgroud of eutrosophc logc Secto 5 presets the role of eutrosophc methods were developed hs paper sheds some lghts o the applcablty of fuzzy set theory ad eutrosophc logc solvg mult-crtera decso makg problems lso t presets the possble applcatos of each method MCDM dfferet felds logc solvg mult-crtera decso makg problems ally coclusos ad potetal future scope of research are descrbed Cocluso secto uzzy Set heory uzzy set theory was frst troduced 965 by Zadeh [5] t s a eteso of classcal set theory that helps solvg problems wth ucerta data ad hadlg formato epressed vague ad mprecse terms [6] ts great stregth appears hadlg mprecse put ad problems wth great complety; however fuzzy systems are cosdered dffcult ad comple to develop ad may cases they may requre umerous smulatos before beg used the real world [7] uzzy set theory s establshed ad has bee used may applcatos such as egeerg ecoomcs evrometal ad socal sceces medce ad maagemet [7] Zadeh [5] troduced may deftos of fuzzy sets such as: Let X be a space of pots wth a geerc elemet of X deoted by hus X = {} fuzzy set X s characterzed by a membershp fucto f whch assocates wth each pot X a real umber the terval [0] wth the values of f at represetg the "grade of membershp" of hus the earer the value of f to uty the hgher the grade of membershp of fuzzy umber s a fuzzy subset the uverse of dscourse X whose membershp fucto s both Cove ad ormal [8] fuzzy set s defed by a membershp fucto used to map a tem oto a terval [0 ] that ca be assocated wth lgustc terms [9] tragu- bdel asser H Zaed ad Hagar M agub pplcatos of uzzy Set heory ad eutrosophc Logc Solvg Mult-crtera Decso Makg Problems

40 eutrosophc Sets ad Systems Vol lar fuzzy umber s a specal case of a trapezodal fuzzy umber ad t s a very popular ad commo tool fuzzy applcatos [0] gure : shows a fuzzy umber [5] 3 pplcatos of uzzy set MCDM 3 Software Selecto eld Se et al [] proposed a mult crtera decso makg approach for Eterprse Resource Plag ERP software selecto usg a heurstc algorthm a fuzzy mult-crtera ad a mult objectve programmg model he proposed approach amed to evaluate the fuctoal ad o-fuctoal ERP software characterstcs o valdate the approach the researchers appled t o a electroc compay urkey ad the results were satsfyg for the compay s decso makers he researchers recommeded combg ther method wth epert system for future work L et al [] frst developed some aggregato operators for aggregatg hestat fuzzy lgustc formato: hestat fuzzy lgustc weghted average HLW operator hestat fuzzy lgustc ordered weghted average HLOW operator ad hestat fuzzy lgustc hybrd average HLH operator the the researchers used these operators fuzzy approaches for solvg ERP software selecto problem he proposed method was appled o a real world case study ad t esured ts capablty selectg the best ERP software that suted the orgazato eeds Ozturkoglu ad Esedemr [3] combed the power of grey relatoal aalyss GR wth a tutostc fuzzy set S mult-crtera method for developg a hybrd ERP software selecto model fter makg a survey of all crtera affectg the ERP software selecto process ad the software packages alteratves the researchers used the S method for obtag the weght of each crtera the the GR method was used for rakg the alteratves ad selectg the best oe servce provder frm whch offered trasportato warehousg ad packagg servces was used as a case study ad the model helped the frm to select the most sutable ERP package Vahd et al [4] used the fuzzy logc for developg a model for ERP software selecto tragular fuzzy membershp fucto was used for processg each crtero to measure the effcecy level of each ERP system alteratve or future work the researchers suggested usg a method based o daptve-etwork-based uzzy ferece Systems S as S method used a learg algorthm that smulate a gve trag data set Le ad Cha [5] developed a uzzy-alytc Herarchy Process -HP ERP software selecto model he proposed model was used two case studes: a compay ad a college for selectg the best ERP software that mate ther eeds Cebec [6] preseted a approach for selectg the best ERP system tetle dustry by usg the balaced scorecard ad uzzy-hp method he ams of ths research were usg balaced scorecard for defg the busess objectves ad matchg them wth ERP packages capabltes ad usg uzzy-hp model for rakg ad selectg the most sutable ERP software package Out ad Efedgl [7] troduced a uzzy-hp model for helpg orgazatos selectg ERP software the presece of vagueess ad wth cosderato to cost ad qualty crtera he researchers combed uzzy method to the HP model to solve the problems of ambgutes ad vagueess accompaed by software selecto problem t the ed of the research a real world case study was solved usg the proposed model ad a comparso betwee HP ad uzzy-hp solutos was coducted ad the results cluded that uzzy-hp method showed more accurate results ad fleblty addg ew ERP software selecto crtera Demrtas et al [8] preseted a two stage decso makg model for ERP software selecto process ad appled the model o a urba trasportato compay t the frst stage by usg uzzy-hp model the model helped the compay to frst take the decso whether t would develop a ew software package or t would use a vedor software package f the decso was usg a vedor software package the movg to the secod stage by usg uzzy-echque for Order Preferece by Smlarty to deal Soluto OPSS model the model helped the compay to select the most sutable software package fttg ts eeds ad epectato Kara ad Chekhrouhou [9] proposed a four steps decso makg methodology for selectg busess maagemet system to Small ad Medum szed Eterprses rst the selecto crtera were collected ad determed by eperts the crtera weghts were calculated usg uzzy-hp combed to OPSS fally the best alteratve was selected or esurg the methodology effectveess a sestvty aalyss was coducted ad the results demostrated that ucertaty was reduced Klc et al [0] used the stregth of uzzy-hp ad OPSS mult-crtera decso makg methods for devel- bdel asser H Zaed ad Hagar M agub pplcatos of uzzy Set heory ad eutrosophc Logc Solvg Mult-crtera Decso Makg Problems

41 eutrosophc Sets ad Systems Vol opg a three stage hybrd model for ERP system selecto ad appled the model for the rle dustry he frst model stage was the determato of all ERP selecto process factors ad crtera ad detfyg ERP software packages as alteratves the secod stage was usg the uzzy-hp method for obtag weghts for each decso crtera the fal model stage was usg the OPSS method for rakg the alteratves ad selectg the best oe he researchers used the proposed model for helpg the urksh rles selectg ERP software package for ts mateace ceter ad the model proved ts effectveess ad effcecy Volarc et al [] proposed a uzzy HP-OPSS model for selectg the best multmeda software for learg ad teachg purposes he uzzy HP method was used for assgg the weght of each crtero ad demostratg the beeft of each crtero to aother fally the OPSS method was used for rakg the multmeda software systems ad selectg the best oe Efe [] developed a hybrd model by tegratg uzzy-hp ad uzzy-opss for ERP software selecto rst the selecto crtera were determed the the weght of each crtero was determed usg uzzy-hp after that uzzy-opss was used for choosg the most approprate ERP software alteratve or esurg the model effectveess t was appled o a electroc frm ad the results demostrated that the model decreased the ucertaty ad the formato loss group decso makg or future work the researcher recommeded usg type fuzzy MCDM methods the ERP selecto process Karsak ad Ozogul [3] developed a mult-crtera decso framework usg o qualty fucto deploymet QD fuzzy lear regresso ad zero oe goal programmg for ERP software selecto he QD method was used for determg ad establshg the relatoshps betwee user demads ad software characterstcs whle the fuzzy lear regresso method was used for assgg values to the ERP software characterstcs ad fally the zero oe goal programmg was used for determg the ERP software alteratve that acheve the mamum values of compay eeds he proposed model was appled o a urksh automotve parts maufacturer to esure ts effectveess 3 Rsk ssessmet ad Success actors Evaluato Je et al [4] troduced a tegrated fuzzy etropyweght MCDM method ad appled t to evaluate ad assess rsk of hydropower statos the Xag Rver Shafee [5] used uzzy alytc etwork Process - P approach based o Chag s etet aalyss for selectg the most approprate rsk mtgato strategy for offshore wd farms Kog ad Lu [6] combed uzzy sets wth HP for developg a MCD model to evaluate success factors E-commerce projects order to help the decso makers to determe ew opportutes for ther orgazatos 33 Ste Selecto eld Rezaeya et al [7] used uzzy-p for selectg the approprate locato of greehouses Mazadara provce ra he applcato of the model esured ts effcecy the selecto process ad rakg of alteratves Vahda et al [8] used uzzy-hp hosptal ste selecto ad determg the optmum ste for a ew hosptal the ehra urba area Chou et al [9] developed a MCDM model by combg uzzy set theory ad smple addtve weghtg SW to evaluate faclty locatos alteratves ad selectg the best oe 34 Suppler Selecto eld Kahrama et al [30] proposed a uzzy-hp model for suppler selecto the researchers determed the selecto crtera ad the the model was used to select the most sutable suppler that mate the compay eeds yha [3] preseted a uzzy-hp model for helpg the frms to select the best suppler accordg to the frm selecto crtera ad for esurg the model effectveess t was apples o a gear motor compay for assessg ts supplers ad selectg the best oe Juor et al [3] proposed a comparatve aalyss of uzzy-hp ad uzzy-opss solvg the problem of suppler selecto oth methods were appled o a trasmsso cables for motorcycles maufacturer whch eeded to select the sutable suppler amog fve alteratves ad based o fve selecto crtera ad the results showed that both methods were helpful however the uzzy-opss method was more effectve the suppler selecto problem Darga et al [33] developed a mult-crtera decso makg framework for helpg the raa automotve dustry suppler selecto process rst the researchers made a huge survey for determg the most crtcal factor the suppler selecto process by usg the omated Group echque G ad the result was seve crtcal factors a uzzy alytcal etwork Process -P was the used for determg weghts of each selecto factor ad selectg the most approprate suppler the model was appled o a automotve compay ad t esured ts effectveess Gupta et al [34] developed a tegrated uzzy HP - uzzy Preferece Rakg Orgazato Method for Erchmet Evaluatos PROMEHEE model for servce provder selecto uder coflctg crtera ad ucertaty evromet rst the selecto crtera were determed the uzzy HP method was used for calculatg the weght of each crtero after that uzzy PROME- HEE method was used for selectg the best alteratve that suted the orgazato eeds Geometrcal alyss for teractve d G software was the used for demostratg the model results ad provdg better uderstadg a sestvty aalyss was coducted to esure the model valdty ad model results esured hgh sestv- bdel asser H Zaed ad Hagar M agub pplcatos of uzzy Set heory ad eutrosophc Logc Solvg Mult-crtera Decso Makg Problems

42 eutrosophc Sets ad Systems Vol ty to chage crtera weghts fally the proposed model was appled o a real world case study a cermet compay to select the most approprate servce provder ad the model esured ts effectveess Haleh ad Hamd [35] used fuzzy sets to assess ad rak the supplers ad selectg the best oe 35 Outsourcg Selecto eld Kahrama et al [36] tred to solve the selecto problem of the rght ERP outsourcg alteratves uder ucertaty codtos usg uzzy-hp mult-crtera decso makg method the researchers appled the proposed model o a automotve frm to help t select the best ERP outsourcg alteratve ad the model proved ts effectveess Che et al [37] tegrated the tragular fuzzy method wth PROMEHEE method for selectg the most approprate outsourcg parter for orgazatos based o seve selecto crtera ad the proposed model was appled o a real world case study ad helped the orgazato to select the most sutable outsourcg parter amog four alteratves 36 Other MCDM elds Ylmaz ad Dagdevre [38] tegrated uzzy- PROMEHEE method wth zero-oe goal programmg to develop a MCD approach for equpmet selecto amog coflctg crtera or hadlg the ucertaty problem wth the qualty maagemet cosultat selecto process Kabr ad Sum [39] used fuzzy set theory as t s a powerful tool for hadlg ucertaty therefore uzzy method was tegrated wth the HP method for determg the selecto crtera weghts the the PROMEHEE method was used for assocatg a preferece fucto to each crtero ad rakg the alteratves or etedg the power of Data evelopmet aalyss DE MCDM method We ad L [40] troduced a uzzy-de method for rakg all the decso-makg uts DMUs for solvg the fuzzy model a hybrd algorthm combed wth fuzzy smulato ad geetc algorthm was used fally a umercal eample was used for llustratg how the model worked Yue ad g [4] tegrated the tragular fuzzy umber ad rakg method wth PROMEHEE method for developg a hybrd model used tet book selecto ad the model was appled o a case study to esure ts valdty ad effectveess 4 eutrosophc Logc realstc decso makg stuatos formato caot always be descrbed by uque crsp umbers they may mply determacy ad therefore eutrosophy was orgally troduced by Smaradache [4] eutrosophy s a brach of phlosophy whch studes the org ature ad scope of eutraltes ad ther teractos wth dfferet deatoal spectra [4] eutrosophy studes the deas ad otos that are eutral determate vague uclear ambguous ad complete [43] eutrosophc sets are capable of dealg wth ucertaty determate ad cosstet formato therefore Smaradache seek to publsh the cocept of eutrosophc set all sceces braches socal sceces ad humates [43] Smaradache refed the eutrosophc set to compoets: t t ; k ; f f fl wth j+k+l = > 3 [43] he basc cocept of eutrosophc set s a geeralzato of classcal set or crsp set [44 45] fuzzy set [5] tutostc fuzzy set [46] fter Smaradache s troducg the cocept of eutrosophc set dfferet sets were quckly proposed the lterature Wag et al [47] eteded the cocept of eutrosophc set to sgle valued eutrosophc sets SVSs ad they also studed the set theoretc operators ad varous propertes of SVSs; may other sets were troduced such as eutrosophc soft set [48] weghted eutrosophc soft sets [49] geeralzed eutrosophc soft set [50] eutrosophc parametrzed soft set [5] eutrosophc soft epert sets [5 53] eutrosophc soft mult-set [54] eutrosophc bpolar set [55] eutrosophc cubc set [56 57] rough eutrosophc set [58 59] terval rough eutrosophc set [60] terval-valued eutrosophc soft rough sets [6 6] etc 5 pplcatos of eutrosophc Logc MCDM Yag ad L [63] proposed ew aggregato operators uder sgle-valued eutrosophc evromet he researchers used sgle-valued eutrosophc set SVS whch s a eteso of tradtoal fuzzy set as SVS ca hadle complete ad cosstet formato the a MCDM method was troduced accordg to the proposed operators ad cose smlarty measures fally the proposed method was appled o a llustratve eample of helpg a vestmet compay to select the best vestmet opto ad the results demostrated that the proposed method was practcal ad effectve or future work the researchers recommeded studyg ew aggregato operators uder eutrosophc evromet Jecy ad rockara [64] proposed a model based o adjustable ad mea potetalty approach by meas of sgle valued eutrosophc level soft sets also the oto of weghted sgle valued eutrosophc soft set was troduced wth a vestgato to ts applcablty decso makg a mprecse evromet swas et al [65] proposed a method wth the am of dealg wth mprecseess ad completeess formato of decso maker s assessmets to acheve better soluto to mult-crtera decso makg problems he researcher troduced tragular fuzzy umber eutrosoph- bdel asser H Zaed ad Hagar M agub pplcatos of uzzy Set heory ad eutrosophc Logc Solvg Mult-crtera Decso Makg Problems

43 eutrosophc Sets ad Systems Vol c sets by tegratg tragular fuzzy umbers wth sgle valued eutrosophc set or esurg the proposed method effectveess t was used to help a medcal frm selectg a medcal represetatve Ch ad Lu [66] troduced a MCDM model by tegratg OPSS method wth terval eutrosophc set for solvg mult-crtera decso makg problems ucertaty evromet he proposed method was used helpg a vestmet compay to select the best vestmet opto ad the results demostrated ts smplcty ad ease of use swas et al [67] preseted a model for solvg MCDM problems wth mssg or ukow formato about crtera weghts hey used Grey Relatoal alyss GR wth sgle-value eautrosophc for developg the model fally a llustratve eample was used to esure model practcalty ad effectveess Dey et al [68] eteded the grey relatoal aalyss GR problems wth terval eutrosophc for solvg MCDM problems wth complete or ukow weghts of crtera he researchers frst developed two optmzato models for recogzg crtera weghts the eteded GR was used for rakg the alteratves fally a umercal eample was used to esure the applcablty of the method roum et al [69] proposed a eteded OPSS model for solvg MCDM problems OPSS was tegrated wth terval eutrosophc for ts great ablty hadlg cosstet formato he eteded OPSS model used terval eutrosophc for represetg the values of the crtera the alteratves were raked usg OPSS method ally a eample was solved to llustrate the model effectveess or solvg ucerta mprecse complete ad cosstet formato MCDM problems Zhag ad Wu [70] developed a two-stage method for sgle-valued eutrosophc or terval eutrosophc mult-crtera decso makg rst a mamzg devato method was troduced for assgg crtera weghts uder terval eutrosophc evromets the OPSS was used for rakg the alteratves ad selectg the optmum choce ally the method was appled a real world case study ad proved ts effectveess Che ad Ye [7] troduced a projecto model of eutrosophc umbers ad ts applcato for solvg the MCDM problem of clay-brcks selecto a actual case was used for applyg the model ad the results demostrated model s applcablty ad ease of use Ye [7] developed a sgle valued eutrosophc crossetropy measure ad ts MCDM method was proposed based o the proposed cross etropy uder sgle valued eutrosophc evromet ally a llustratve eample was solved to llustrate the applcato of the proposed method Pramak ad Modal [73] troduced a MCDM method based o terval eutrosophc sets where the ratg of alteatves was epressed wth terval eutrosophc values characterzed by terval truthmembershp degree terval determacy-membershp degree ad terval falsty-membershp degree he sgle valued eutrosophc grey relatoal aalyss method was eteded to terval eutrosophc evromet ad appled MCDM problems ally a llustratve eample was solved to llustrate the applcato of the proposed method Madal ad asu [74] developed a ew smlarty measures eutrosophc evromet based o hypercompe umber system for rakg alteratves ad selectg the best oe whle solvg MCDM problems ally a umercal eample was troduced to esure the method effectveess Modal ad Pramak [75] troduced a MCDM method based o Dce ad Jaccard smlarty measures of terval rough eutrosophc set ad terval eutrosophc mea operator ad fally they appled the method o a laptop selecto case swas et al [76] troduced cose smlarty measure betwee two trapezodal fuzzy eutrosophc umbers for solvg MCDM problems uder eutrosophc evromet ad a umercal eample was solved to llustrate the method work Ma et al [77] troduced a tme seres aalyss approach tegrated wth terval eutrosophc sets for selectg trustworthy cloud servce hree umercal eamples were used to llustrate the approach applcablty ad effcecy selectg rsk-sestve servce Modal ad Pramak [78] developed a eutrosophc MCDM model based o hybrd score-accuracy fuctos of sgle valued eutrosophc umbers for teacher selecto recrutmet process hgher educato a llustratve eample was troduced for demostratg the model work Modal ad Pramak [79] also proposed a sgle valued eutrosophc MCDM model for selectg the best school for chldre umercal eample was used to prove the model effcecy Ye ad Smaradache [80] troduced a refed sglevalued eutrosophc set RSVS ad a smlarty measure of RSVSs the a MCDM method usg RSVS formato was preseted based o the smlarty measure of RSVSs ally a real case study was used for applyg the metod to help a costructo frm selectg the best project ad the results demostrated the method effectveess Modal ad Pramak [8] troduced a rough eutrosophc mult-attrbute decso makg method based o grey relatoal aalyss by etedg the eutrosophc grey relatoal aalyss method to rough eutrosophc grey relatoal aalyss method ad applyg t to mult-attrbute decso makg problem ths method the ratg of all alteratves was epressed wth upper ad lower appromato operator ad the par of eutrosophc sets whch were characterzed by truth-membershp degree determacy-membershp degree ad falstymembershp degree ally a umercal eample was used bdel asser H Zaed ad Hagar M agub pplcatos of uzzy Set heory ad eutrosophc Logc Solvg Mult-crtera Decso Makg Problems

44 eutrosophc Sets ad Systems Vol to demostrate the method applcablty Modal ad Pramak [8] also proposed a rough eutrosophc mult-attrbute decso makg method based o rough accuracy score fucto he ratg of all alteratves was epressed wth upper ad lower appromato operator ad the par of eutrosophc sets whch were characterzed by truth-membershp degree determacy-membershp degree ad falstymembershp degree ally a umercal eample was used to esure the method effectveess Peg et al [83] troduced a ew outrakg approach for solvg MCDM problems uder eutrosophc evromet by tegratg smplfed eutrosophc sets wth ELECRE method wo practcal eamples were provded to esure the practcalty ad effectveess of the proposed approach Ye [84] troduced a ew MCDM method usg the weghted correlato coeffcet or the weghted cose smlarty measure of sgle-valued eutrosophc sets where the alteratves evaluato was made by truthmembershp degree determacy-membershp degree ad falsty-membershp degree uder sgle-valued eutrosophc evromet ally a eample was solved for provg the applcablty of the proposed method swas et al [85] proposed a rakg method for solvg MCDM problems usg sgle-valued trapezodal eutrosophc umbers SVrs whch was a specal case of sgle-valued eutrosophc umbers ally a eample was used for demostratg the model effcecy Cocluso hs study demostrated the role of fuzzy set theory ad eutrosophc logc the feld of mult-crtera decso makg; applcatos ad researches of the two methods were preseted to llustrate the mprovemets ad developmets made MCDM feld usg those two methods t s cocluded that there s a weakess pot eutrosophc sets applcatos MCDM real world case studes lthough there are may researchers that use umercal eamples for applyg the eutrosophc model there s a shortage real case studes usage lso eutrosophc logc should be appled more MCDM felds lke suppler selecto software selecto rsk assessmet ad other felds where fuzzy set theory made a otceable developmet to vestgate ts stregth ad weakess pots herefore there are may future works that ca be doe such as: pply eutrosophc logc o dfferet decso support problems pply eutrosophc logc o software egeerg 3 Propose ew adaptve mechasm to update eutrosophc logc 4 Solve tme seres forecastg 5 alyze the effect of hybrdzg eutrosophc logc wth meta-heurstcs algorthms 6 pply eutrosophc logc wth eural etworks 7 Desg eutrosophc logc Cotroller by Partcle Swarm Optmzato Refereces [] D Staujkc Dordevc ad M Dordevc Comparatve aalyss of some promet MCDM methods: case of rakg Serba baks Serba Joural of Maagemet [] G Salvatore Multple Crtera Decso alyss: State of the rt Survey teratoal Seres Operatos Research & Maagemet Scece Vol p 048 [3] S Levta E drews Glsea J erguso R oel ad PM Copla pplcato of the R framework to case studes: observatos ad sghts Cl Pharmacol her [4] J Devl ad J Susse corporatg multple crtera H: methods ad processes Lodo: he Offce of Health Ecoomcs 0 [5] L Zadeh uzzy sets formato ad Cotrol [6] J almat Lafot R Mafret ad Pessel decsomakg system to martme rsk assessmet Ocea Egeerg [7] M Velasquez ad P Hester alyss of Mult-Crtera Decso Makg Methods teratoal Joural of Operatos Research Vol 0 o [8] Kaufma ad M M Gupta troducto to fuzzy arthmetc: theory ad applcatos ew York: Va ostrad Rehold Compay 985 [9] Y-C Lee -P Hog ad -C Wag Mult-level fuzzy mg wth multple mmum supports Epert Systems wth pplcatos [0] Y-H Che -C Wag ad C-Y Wu Strategc decsos usg the fuzzy PROMEHEE for S outsourcg Epert Systems wth pplcatos [] C G Se H arach S Se ad H aslıgl tegrated decso support system dealg wth qualtatve ad quattatve objectves for eterprse software selecto Epert Systems wth pplcatos [] R L X Zhao ad G We Models for selectg a ERP system wth hestat fuzzy lgustc formato Joural of tellget & uzzy Systems DO:0333/S [3] Y Ozturkoglu ad E Esedemr ERP Software Selecto usg S ad GR Methods Joural of Emergg reds Computg ad formato Sceces [4] J Vahd D D SalooKolay ad Yavar Model for Selectg a ERP System wth ragular uzzy umbers ad Mamda ferece J Math Computer Sc [5] C- Le ad H-L Cha Selecto Model for ERP System by pplyg uzzy HP pproach teratoal Joural of the Computer the teret ad Maagemet Vol 5 o3 007 pp 58-7 [6] U Cebec uzzy HP-based decso support system for selectg ERP systems tetle dustry by usg balaced bdel asser H Zaed ad Hagar M agub pplcatos of uzzy Set heory ad eutrosophc Logc Solvg Mult-crtera Decso Makg Problems

45 eutrosophc Sets ad Systems Vol scorecard Epert Systems wth pplcatos [7] S Out ad Efedgl theorcal model desg for ERP software selecto process uder the costrats of cost ad qualty: fuzzy approach Joural of tellget & uzzy Systems DO:0333/S [8] Demrtas O lp U R uzkaya ad H araçl uzzy HP-OPSS two stages methodology for ERP software selecto: applcato passeger trasport sector 5th teratoal Research/Epert Coferece reds the Developmet of Machery ad ssocated echology 0 Prague Czech Republc -8 [9] S S Kara ad Chekhrouhou Mult-crtera group decso makg approach for collaboratve software selecto problem Joural of tellget ad uzzy Systems 03 DO:0333/S-073 [0] H S Klc S Zam ad D Dele Developmet of a hybrd methodology for ERP system selecto: he case of urksh rles Decso Support Systems [] Volarc E rajkovc ad Sjekavca tegrato of HP ad OPSS Methods for the Selecto of pproprate Multmeda pplcato for Learg ad eachg teratoal joural of mathematcal models ad methods appled sceces Vol [] Efe tegrated uzzy Mult Crtera Group Decso Makg pproach for ERP System Selecto ppled Soft Computg Joural 05 [3] E E Karsak ad C O Ozogul tegrated decso makg approach for ERP system selecto Epert Systems wth pplcatos [4] Y J G H Huag ad W Su Rsk assessmet of hydropower statos through a tegrated fuzzy etropy-weght multple crtera decso makg method: case study of the Xag Rver Epert Systems wth pplcatos [5] M Shafee fuzzy aalytc etwork process model to mtgate the rsks assocated wth offshore wd farms Epert Systems wth pplcatos [6] Kog ad H Lu pplyg uzzy alytc Herarchy Process to evaluate success factors of E-commerce 005 teratoal joural of formato & systems sceces Vol [7] Rezaeya S Ghadkolae J Mehr-ekmeh ad H R Rezaeya uzzy P pproach for ew pplcato: Greehouse Locato Selecto; a Case ra J Math Computer Sc [8] M H Vahda leshekh ad lmohammad Hosptal ste selecto usg fuzzy HP ad ts dervatves Joural of Evrometal Maagemet [9] S-Y Chou Y-H Chag ad C-Y She fuzzy smple addtve weghtg system uder group decso-makg for faclty locato selecto wth objectve/subjectve attrbutes Europea Joural of Operatoal Research [30] C Kahrama U Cebec ad Z Uluka Mult-crtera suppler selecto usg fuzzy HP Logstcs formato Maagemet Vol [3] M YH uzzy HP approach for suppler selecto problem: case study a gearmotor compay teratoal Joural of Maagg Value ad Supply Chas JMVSC Vol4 o DO: 05/jmvsc03430 [3] R L Juor L Osro ad L C R Carpett comparso betwee uzzy HP ad uzzy OPSS methods to suppler selecto ppled Soft Computg [33] Darga jomshoaea M R Galakasha Memara ad M M apa Suppler Selecto: uzzy-p pproach Proceda Computer Scece [34] R Gupta Sachdeva ad hardwaj Selecto of logstc servce provder usg fuzzy PROMEHEE for a cemet dustry Joural of Maufacturg echology Maagemet Vol 3 0 ss [35] H Haleh ad Hamd fuzzy MCDM model for allocatg orders to supplers a supply cha uder Ucertaty over a mult-perod tme horzo Epert Systems wth pplcatos [36] C Kahrama eskese ad Kaya Selecto amog ERP outsourcg alteratves usg a fuzzy mult-crtera decso makg methodology teratoal Joural of Producto Research Vol 48 o [37] Y-H Che -C Wag ad C-Y Wu Strategc decsos usg the fuzzy PROMEHEE for S outsourcg Epert Systems wth pplcatos [38] Ylmaz ad M Dagdevre combed approach for equpmet selecto: -PROMEHEE method ad zero oe goal programmg Epert Systems wth pplcatos [39] G Kabr ad R S Sum tegratg fuzzy aalytc herarchy process wth PROMEHEE method for total qualty maagemet cosultat selecto Producto & Maufacturg Research Vol o [40] M We ad H L uzzy data evelopmet aalyss DE: Model ad rakg method Joural of Computatoal ad ppled Mathematcs [4] K K Yue ad O g etbook Selecto Usg uzzy PROMEHEE Method teratoal Joural of uture Computer ad Commucato Vol o 0 [4] Smaradache ufyg feld logcs eutrosophy: eutrosophc probablty set ad logc merca Research Press Rehoboth 999 [43] Modal ad S Pramak Decso Makg ased o Some smlarty Measures uder terval Rough eutrosophc Evromet eutrosophc Sets ad Systems Vol [44] H J S Smth O the tegrato of dscotuous fuctos Proceedgs of the Lodo Mathematcal Socety Seres [45] G Cator Ü uedlche ad l Puktmagfaltgkete V [O fte lear pot-mafolds sets] Mathematsche ale [46] K taassov tutostc fuzzy sets uzzy Sets ad Systems [47] H Wag Smaradache Y Q Zhag ad R Suderrama Sgle valued eutrosophc sets Multspace ad Multstructure [48] P K Maj eutrosophc soft set als of uzzy Mathematcs ad formatcs bdel asser H Zaed ad Hagar M agub pplcatos of uzzy Set heory ad eutrosophc Logc Solvg Mult-crtera Decso Makg Problems

46 eutrosophc Sets ad Systems Vol [49] P K Maj Weghted eutrosophc soft sets approach a mult-crtera decso makg problem Joural of ew heory [50] S roum Geeralzed eutrosophc soft set teratoal Joural of Computer Scece Egeerg ad formato echology [5] S roum Del ad Smaradache eutrosophc parametrzed soft set theory ad ts decso makg teratoal roter Scece Letters 04-0 [5] M Şah S lkhazaleh ad V Uluçay eutrosophc soft epert sets ppled Mathematcs [53] S roum ad Smaradache Sgle valued eutrosophc soft epert sets ad ther applcato decso makg Joural of ew heory [54] Del S roum ad M l eutrosophc soft mult-set theory ad ts decso makg eutrosophc Sets ad Systems [55] Del M l Smaradache polar eutrosophc sets ad ther applcato based o mult-crtera decso makg problems 05 [56] M l S roum ad Smaradache he theory of eutrosophc cubc sets ad ther applcato patter recogto Joural of tellget ad uzzy System 05 [57] Y Ju Smaradache ad CS Km eutrosophc cubc sets JSK-500R [58] S roum Smaradache ad M Dhar Rough eutrosophc sets tala joural of pure ad appled mathematcs [59] S roum Smaradache ad M Dhar Rough eutrosophc sets eutrosophc Sets ad Systems [60] S roum ad Smaradache terval eutrosophc rough sets eutrosophc Sets ad Systems [6] S roum ad Smaradache Soft terval valued eutrosophc rough sets eutrosophc Sets ad Systems [6] S roum ad Smaradache terval-valued eutrosophc soft rough sets teratoal Joural of Computatoal Mathematcs 05 [63] L Yag ad L Mult-Crtera Decso-Makg Method Usg Power ggregato Operators for Sgle-valued eutrosophc Sets teratoal Joural of Database ad heory ad pplcato Vol9 o 06 pp3-3 [64] J M Jecy ad rockara djustable ad Mea Potetalty pproach o Decso Makg eutrosophc Sets ad Systems Vol 06-0 [65] P swas S Pramak ad C Gr ggregato of tragular fuzzy eutrosophc set formato ad ts applcato to mult-attrbute decso makg eutrosophc Sets ad Systems Vol [66] P Ch ad P Lu eteded OPSS method for the multple attrbute decso makg problems based o terval eutrosophc set eutrosophc Sets ad Systems Vol 03-8 [67] P swas S Pramak ad C Gr ew Methodology for eutrosophc Mult-ttrbute Decso-Makg wth Ukow Weght formato eutrosophc Sets ad Systems Vol [68] P P Dey S Pramak ad C Gr eteded grey relatoal aalyss based multple attrbute decso makg terval eutrosophc ucerta lgustc settg eutrosophc Sets ad Systems Vol [69] S roum J Ye ad Smaradache Eteded OPSS Method for Multple ttrbute Decso Makg based o terval eutrosophc Ucerta Lgustc Varables eutrosophc Sets ad Systems Vol [70] Z Zhag ad C Wu ovel method for sgle-valued eutrosophc mult-crtera decso makg wth complete weght formato eutrosophc Sets ad Systems Vol [7] J Che ad J Ye Projecto Model of eutrosophc umbers for Multple ttrbute Decso Makg of Clay-rck Selecto eutrosophc Sets ad Systems Vol [7] J Ye Sgle valued eutrosophc cross-etropy for multcrtera decso makg problems ppl Math Modellg 03 do: [73] S Pramak ad K Modal terval eutrosophc Mult- ttrbute Decso-Makg ased o Grey Relatoal alyss eutrosophc Sets ad Systems Vol [74] K Madal ad K asu Hypercomple eutrosophc Smlarty Measure & ts pplcato Multcrtera Decso Makg Problem eutrosophc Sets ad Systems Vol [75] K Modal ad S Pramak Decso Makg ased o Some smlarty Measures uder terval Rough eutrosophc Evromet eutrosophc Sets ad Systems Vol [76] P swas S Pramak ad C Gr Cose Smlarty Measure ased Mult-attrbute Decso-makg wth rapezodal uzzy eutrosophc umbers eutrosophc Sets ad Systems Vol [77] H Ma Z Hu K L ad H Zhag oward trustworthy cloud servce selecto: tme-aware approach usg terval eutrosophc set J Parallel Dstrb Comput 06 [78] K Modal ad S Pramak Mult-crtera Group Decso Makg pproach for eacher Recrutmet Hgher Educato uder Smplfed eutrosophc Evromet eutrosophc Sets ad Systems Vol [79] K Modal ad S Pramak eutrosophc Decso Makg Model of School Choce eutrosophc Sets ad Systems Vol [80] J Ye ad Smaradache Smlarty Measure of Refed Sgle-Valued eutrosophc Sets ad ts Multcrtera Decso Makg Method eutrosophc Sets ad Systems Vol [8] K Modal ad S Pramak Rough eutrosophc Mult- ttrbute Decso-Makg ased o Grey Relatoal alyss eutrosophc Sets ad Systems Vol [8] K Modal ad S Pramak Rough eutrosophc Mult- ttrbute Decso-Makg ased o Rough ccuracy Score ucto eutrosophc Sets ad Systems Vol [83] J-J Peg J-Q Wag H-Y Zhag ad X-H Che outrakg approach for mult-crtera decso-makg problems bdel asser H Zaed ad Hagar M agub pplcatos of uzzy Set heory ad eutrosophc Logc Solvg Mult-crtera Decso Makg Problems

47 eutrosophc Sets ad Systems Vol wth smplfed eutrosophc sets ppled Soft Computg 04 [84] J Ye Multcrtera decso-makg method usg the correlato coeffcet uder sgle-valued eutrosophc evromet teratoal Joural of Geeral Systems Vol 4 o [85] P swas S Pramak ad C Gr Value ad ambguty de based rakg method of sgle-valued trapezodal eutrosophc umbers ad ts applcato to mult-attrbute decso makg eutrosophc Sets ad Systems Vol Receved: October 0 06 ccepted: December 5 06 bdel asser H Zaed ad Hagar M agub pplcatos of uzzy Set heory ad eutrosophc Logc Solvg Mult-crtera Decso Makg Problems

48 eutrosophc Sets ad Systems Vol Uversty of ew Meco rregular eutrosophc Graphs asr Shah ad Sad roum Departmet of Mathematcs Rphah teratoal Uversty -4 slamabad Paksta Emal: memaths@yahoocom Laboratory of formato processg aculty of scece e M'Sk Uversty Hassa P 7955 Sd Othma Casablaca Morocco Emal: broumsad78@gmalcom bstract he cocepts of eghbourly rregular eutrosophc graphs eghbourly totally rregular eutrosophc graphs hghly rregular eutrosophc graphs ad hghly totally rregular eutrosophc graphs are troduced crtera for eghbourly rregular ad hghly rregular eutrosophc graphs to be equvalet s dscussed Keywords: eutrosophc graphs rregular eutrosophc graphs eghbourly rregular eutrosophc graphs Hghly rregular eutrosophc graphs troducto zrel Rosefeld [6] troduced the oto of fuzzy graphs 975 whch have may applcatos modelg Evrometal sceces socal sceces Geography ad Lgustcs Some remarks o fuzzy graphs are gve by P hattacharya [4] J Mordeso ad C S Peg defed dfferet operatos o fuzzy graphs hs paper [0] he cocept of bpolar fuzzy sets was tated by Zhag [4] bpolar fuzzy set s a eteso of the fuzzy set whch has a par of postve ad egatve membershp values ragg [ ] usual fuzzy sets the membershp degrees of elemets rage over the terval [ 0] he membershp degree epresses the degree of beloggess of elemets to a fuzzy set he membershp degree dcates that a elemet completely belogs to ts correspodg fuzzy set ad the membershp degree 0 dcates that a elemet does ot belog to the fuzzy set he membershp degrees o the terval 0 dcate the partal membershp to the fuzzy set Sometmes the membershp degree meas the satsfacto degree of elemets to some property or costrat correspodg to a fuzzy set polar fuzzy sets membershp degree rage s elarged from the terval [ 0] to [ ] a bpolar valued fuzzy set the membershp degree 0 dcate that elemets are rrelevat to the correspodg property the membershp degrees o 0] shows that elemets some what satsfy the property ad the membershp degrees o [0 shows that elemets somewhat satsfy the mplct couter-property may domas t s mportat to be able to deal wth bpolar formato t s oted that postve formato represets what s grated to be possble whle egatve formato represets what s cosdered to be mpossble he frst defto of bpolar fuzzy graphs was troduced by kram []whch are the etesos of fuzzy graphs He defed dfferet operatos of uo tersecto complemet somorphsms hs paper Smaradache [] troduced oto of eutrosophc set whch s useful for dealg real lfe problems havg mprecse determacy ad cosstet data he theory s geeralzato of classcal sets ad fuzzy sets ad s appled decso makg problems cotrol theory medces topology ad may more real lfe problems Shah ad Hussa troduced the oto of soft eutrosophc graphs [7] Shah troduced the oto of eutrosophc graphs ad dfferet operatos lke uo tersecto complemet hs work [8] urthermore he defed dfferet morphsms o eutrosophc graphs ad proved related theorems the preset paper the cocepts of eghbourly rregular eutrosophc graphs eghbourly totally rregular eutrosophc graphs hghly rregular eutrosophc graphs hghly totally rregular eutrosophc graphs ad eutrosophc dgraphs are troduced Some results o rregularty of eutrosophc graphs are also prove secto some basc cocepts about graphs ad eutrosophc sets are gve Secto3 s about eutrosophc graphs ther dfferet operatos ad rregularty of eutrosophc graphs Eamples alog wth fgures are also gve to make the deas clear PRLMRES ths secto we have gve some deftos about graphs ad eutrosophc sets hese wll be helpful later sectos Defto [] graph G cossts of set of fte objects V v v v 3 v called vertces also called pots or odes ad other set E e e e 3 e whose elemet are called edges also called les or arcs Usually a graph s deoted as G VE Let G be a graph ad uv a edge of G Sce uv s - elemet set we may wrte vu stead of uv t s ofte more coveet to represet ths edge by uv or vu Defto [5] he cardalty of V e the o asr Shah ad Sad roum rregular eutrosophc Graphs

49 eutrosophc Sets ad Systems Vol of vertces s called the order of graph G ad deoted by V he cardalty of E e the umber of edges s called the sze of the graph ad deoted by E Let V G { v v v} ad E G { e e e } be the set of vertces ad edges of a graph G Each edge e k E G s detfed wth a uordered par j v v of vertces he vertces the ed vertces of e k v ad v j are called 3 Defto [5] wo vertces joed wth a edge are called adjacet vertces 4 Defto [5] [0 edge e of a graph G s sad to be cdet wth a verte v ad vce versa f v s the ed verte of e y two o-parallel edges say e ad e j are sad to be adjacet f e ad e j are cdet wth a verte v 5 Defto [5] he degree of ay verte v of G s the umber of edges cdet wth verte v Each selfloop s couted twce Degree of a verte s always a postve umber ad s deoted as degv he mmum degree ad mamum degree of vertces V G are deoted by G ad G respectvely H 6 Defto [5] verte whch s ot cdet wth ay edge s called a solated verte other words a verte wth degree zero s called a solated verte 7 Defto [5] graph wthout self-loops ad parallel edges s called a smple graph 8 Defto [5] smple graph s sad to be regular f all vertces of graph G are of equal degree other words f a graph G G G r e each verte havg degree r the G s sad to be regular of degree r or smply r regular 9 Defto [5] graph G V subgraph of G V E E G E G ad each edge of ed vertces G as G E G V G s called a f V ad G has the same 0 Defto [5] a graph G a fte alteratg sequece of vertces ad edges v e v e e m v k startg ad edg wth vertces such that each edge the sequece s cdet wth the vertces followg ad precedg t s called a walk a walk o edge appears more tha oce however a verte may appear more tha oce Defto [] a multgraph o loop are allowed but more tha oe edge ca jo two vertces these edges are called multple edges or parallel edges ad a graph s called multgraph Defto [] Let G V E ad G V E be two graphs fucto f : V V s called somorphsm f f s oe to oe ad oto for all a b V { a b} E f ad oly f { f a f b} E whe such a fucto ests ad G G are called somorphc graphs ad s wrtte as G G other words two graph G ad G are sad to be somorphc to each other f there s a oe to oe correspodece betwee ther vertces ad betwee edges such that cdece relatoshp s preserved 3 Defto [] eutrosophc set o the uverse of dscourse X s defed as { : X ]0 [ 0 3 Hece we cosder the eutrosophc set whch takes the values from the subset of 0 4 Defto Let { X} ad { X} be two eutrosophc sets o uverse of dscourse X he s called eutrosophc relato o f y m y y m y X where y ma y for all y X eutrosophc relato o X s called symmetrc f y y y y y y for all y X 3 EUROSOPHC GRPHS D RREGULR- Y ths secto we wll study some basc deftos about eutrosophc graphs ad dfferet types of degrees of ver- asr Shah ad Sad roum rregular eutrosophc Graphs

50 eutrosophc Sets ad Systems Vol tces wll be dscussed rregularty of eutrosophc graphs ad related results are also prove ths secto 3 Defto [8] Let G V E be a smple graph ad E V V Let : V [0] deote the truth-membershp determacymembershp ad falsty- membershp of a elemet V ad : E [0] deote the truth-membershp determacy-membershp ad falsty- membershp of a elemet y E y a eutrosophc graph we mea a 3 - tuple G G such that y m{ y} y m{ y ma{ for all y E 3 Defto [8] Let y} y} G V E ad G V E be two smple graphs he uo of two eutrosophc graphs G G ad G G s deoted by G G G G G where the truthmembershp determacy-membershp ad falstymembershp of uo are as follows f V V f V V ma{ } f V V f V V f V V ma{ } f V V f V V f V V m{ } f V V lso y f y E E y y f y E E ma{ y y} f y E y ma{ y m{ y y y y y y f y E y} f y E f y E y} E E f y E f y E E E f y E 33 Defto [8] he tersecto of two eutrosophc graphs G G ad G G s deoted by G G where G G G V V V ad the truth-membershp determacy-membershp ad falstymembershp of tersecto are as follows m{ } m{{ } ma{{ } also y m{ y y} y m{ y y} y ma{ y y} 34 Defto Let G G be a eutrosophc graph he bhd of a verte s defed as where y V : y m y y V : y m y y V : y ma y 35 Defto Let G G be a eutrosophc graph he bhd degree of a verte G defed by E E asr Shah ad Sad roum rregular eutrosophc Graphs

51 eutrosophc Sets ad Systems Vol deg deg y y g y g y deg y g 36 Defto Let G G be a eutrosophc graph he closed bhd degree of a verte s defed as deg deg deg deg where deg deg [ ] deg [ ] [ ] y y y y y y 37 Defto Let G G be a eutrosophc graph he order of eutrosophc graph deoted by OG s defed as O G O G O G O O G O G V V O G G V Where y he sze of a eutrosophc graph G G s deoted by S G ad s defed as he sze of a eutrosophc graph G G s deoted by SG ad s defed as S S G S G S S G S G G y E G S y E y E y G y y where 38 Defto eutrosophc graph G G s called regular f all the vertces have the same ope bhd degree 39 Defto Let G G be a eutrosophc graph f there s a verte whch s adjacet to vertces wth dstct eghborhood degrees the rregular eutrosophc graph 30 Eample Let G V E V 3 ad s called a be a smple graph wth E 3 3 eutrosophc graph G s gve table below ad j 0 j 0 ad j for all j E able gure 0037 Here deg Smlarly 3 asr Shah ad Sad roum rregular eutrosophc Graphs

52 eutrosophc Sets ad Systems Vol deg deg Clearly G s a rregular eutrosophc graph Defto Let G G be a eutrosophc graph f there s a verte whch s adjacet to vertces wth dstct closed eghborhood degrees the s called a totally rregular eutrosophc graph 3 Eample Cosder a eutrosophc graph below wth V E { } gure Here deg Smlarly deg 04080deg deg Clearly G s eghbourly rregular eutrosophc graph Defto coected eutrosophc graph G G s called eghbourly totally rregular eutrosophc graph f every two adjacet vertces of G have dstct closed eghborhood degrees gure deg 0886 deg[ ] 0886 deg[ deg[ 4 ] 09 5 deg[ 3 3 ] ] Clearly G s totally rregular eutrosophc graph 36 Eample eample of eghbourly totally rregular eutrosophc graph G s gve below wth V { E { } } Defto Let G G be a coected eutrosophc graph f every two adjacet vertces of have dstct ope eghborhood degrees the s called eghbourly rregular eutrosophc graph 34 Eample Cosder a eutrosophc graph G below wthv { } E { } gure 4 deg 0955 deg 77 asr Shah ad Sad roum rregular eutrosophc Graphs

53 eutrosophc Sets ad Systems Vol deg[ 3 deg[ 4 ] 399 ] Defto coected eutrosophc graph G G s called hghly rregular eutrosophc graph f every verte of G s adjacet to vertces wth dstct eghborhood degrees ote hghly rregular eutrosophc graph may ot be eghbourly rregular eutrosophc graph 38 Eample rom fgure 5 below t ca be see that a hghly rregular eutrosophc graph may ot be eghbourly rregular eutrosophc graph Here deg 0406 deg 0608 deg deg Clearly every two adjacet vertces have dstct bhd degree but s adjacet to ad 3 havg same degree Hece G s eghbourly rregular eutrosophc graph but ot hghly rregular eutrosophc graph eghbourly rregular eutrosophc graph may ot be a eghbourly totally rregular eutrosophc graph 30 Eample Cosder a eutrosophc graph such that V } { E { } gure Here V whch s adjacet to the vertces 3 6 wth dstct bhd degrees ut deg deg 3 So G s hghly rregular eutrosophc graph but t s ot a eghbourly rregular eghbourly rregular eutrosophc graph may ot be hghly rregular eutrosophc graph 39 Eample Cosder the graph below X deg 4 gure deg deg deg d deg[ ] 0645 deg[ deg[ 3 ] 0846 deg[ ] 0645 ] 0754 We see that deg[ ] deg[ ] Hece G s eghbourly rregular eutrosophc graph but ot a eghbourly totally rregular eutrosophc graph X gure 6 3 v eghbourly totally rregular eutrosophc graph may ot be a eghbourly rregular eutrosophc graph 3 Eample Cosder a eutrosophc graph G such that V { 3 4} E { } asr Shah ad Sad roum rregular eutrosophc Graphs

54 eutrosophc Sets ad Systems Vol deg[ gure 8 asr Shah ad Sad roum rregular eutrosophc Graphs deg[ ] 0955 deg[ 3 ] 88 deg[ ut deg 07 deg deg 3 07 deg ] ] Hece deg deg deg 3 deg 4 So G s eghbourly totally rregular eutrosophc graph but ot a eghbourly rregular eutrosophc graph 3 Proposto Let be a eutrosophc graph he G s hghly rregular eutrosophc graph ad eghbourly rregular eutrosophc graph ff the eghborhood degrees of all the vertces of G are dstct Proof Let G be a eutrosophc graph wth -vertces 3 Suppose G s both hghly rregular ad eghbourly rregular eutrosophc graph We wat to show the eghborhood degrees of all vertces of dstct Let deg p q r are Let the adjacet vertces of are 3 wth bhd degrees p q r q r p q r p respectvely Sce G s hghly rregular so p p p q q r r r 3 q 3 lso p p p p q q q3 q r r r3 r because G s eghbourly Hece the eghborhood degrees of all the vertces of are dstct Coversely suppose that the eghborhood degrees of all the vertces are dstct ow we wat to show that s hghly rregular ad eghbourly rregular eutrosophc graph Let deg p q r gve that p p p p q q q 3 q ad r r r3 r Every two adjacet vertces have dstct eghborhood degrees ad to every verte the adjacet vertces have dstct eghborhood degrees whch completes the proof 33 Proposto Let G be a eutrosophc graph f G s eghbourly rregular eutrosophc graph ad s a costat fucto the G s a eghbourly totally rregular eutrosophc graph Proof Let G be eghbourly rregular eutrosophc graph Let V where ad j are adjacet rregular So Whch s a cotradcto sce q q Cosder p q r p q r p3 q3 r3 p q r deg deg j r k 3 r k 3 j vertces wth dstct eghborhood degrees p q r ad p q r respectvely Let us assume that where j 3 3 j j k k k3 k k k are costats ad k k k [0] herefore deg [ ] deg p k deg deg q k deg deg r k 3 deg j deg j j p k deg j deg j j q k deg j deg j j r k 3 We wat to show deg degjdeg degjdeg degj Suppose that o cotrary deg [ ] deg [ j ] p k p k Whch p p k k 0 p p s a cotradcto because p p Smlarly deg deg j q q k k 0 q q

55 eutrosophc Sets ad Systems Vol Whch s a cotradcto sce r r herefore G s eghbourly totally rregular eutrosophc graph 34 Proposto eutrosophc graph G of G where G s a cycle wth 3 vertces s eghbourly rregular ad hghly rregular ff the truth- membershp determacymembershp ad falsty- membershp values of vertces betwee every par of vertces are all dstct Proof Suppose that truth membershp determacy ad falsty membershp betwee every par of vertces are all dstct Let V ad j k j k j k Whch mples that j k j j j j j j k k k k hat s deg deg j deg k Smlarly we ca show deg deg j deg k deg deg j deg k showg that deg deg j deg k Hece s eghbourly rregular ad hghly rregular eutrosophc graph Coversely suppose that k k s eghbourly rregular ad hghly rregular Let deg p q r 3 Suppose that truthfuless falsty ad determacy of two vertces are same Let V wth he deg deg deg deg deg deg Whch mples deg deg Sce G s a cycle so we have a cotradcto to the fact thatg s eghbourly rregular ad hghly rregular eutrosophc graph Hece the truthmembershp determacy-membershp ad falstymembershp values of vertces betwee every par of vertces are all dstct totally rregular eutrosophc graph ad s a costat fucto the G s a eghbourly rregular eutrosophc graph Proof We suppose G s eghbourly totally rregular eutrosophc graph he by defto the closed eghborhood degree of every two adjacet are dstct Let V where ad j are adjacet j vertces wth dstct degrees p q ad r p q r respectvely Let us assume that j k k j j k3 where k k k3 are costats ad k k k3 [0] ad deg[ ] ] We wat to show deg[ j deg deg j Sce deg[ ] deg j so deg [ ] deg [ ] deg [ ] deg [ ] deg [ ] deg [ ] ow deg [ ] deg [ ] p p p k k 0 p Smlarly deg [ ] deg [ ] q q j j j j k p p k k k j 0 q q deg [ ] deg [ j ] r k3 r k3 r r k3 k3 0 r r Hece the degrees of V are dstct hs s true for every par of adjacet vertces G herefore G s eghbourly rregular eutrosophc graph j Cocluso eutrosophc sets are the geeralzato of the classcal sets ad of the fuzzy sets ad have may applcatos real world problems whe the data s mprecse determat or cosstet ths paper we tated the dea of the rregular eutrosophc graphs ad dscussed dfferet propertes of such graphs We have see how eghbourly rregular ad hghly rregular eutrosophc graphs are equvalet future we wll eted our work to other graph theory areas by usg eutrosophc graphs 35 Proposto Let G be a eutrosophc graph f G s eghbourly asr Shah ad Sad roum rregular eutrosophc Graphs

56 eutrosophc Sets ad Systems Vol Refereces [] M kram polar fuzzy graphs formato Sceces [] M kram polar fuzzy graphs wth applcatos Kowledge-ased Systems [3] M kramw Dudek Regular polar uzzy Graphs eural comp & applc [4] P hattacharya Some Remarks O uzzy Graphs Patter Recogto Letters [5] K R huta Rosefeld uzzy Ed odes uzzy Graphs formato Sceces [6] S roum M alea akkal ad Samaradache Sgle Valued eutrosophc Graph Joural of ew theory [7] S roum Mohamed alea ssa akal ad loret Smaradache O polar Sgle Valued eutrosophc Graphs Joural of ew heoryyear: [8] L Euler Soluto problemats ad geometram stus pertets Commetar cademae Scetarum mperals Petropoltaae [9] Hussa M Shabr lgebrac Structures Of eutrosophc Soft Sets eutrosophc Sets ad Systems [0] J Mordeso ad C S Peg Operatos O uzzy Graphs formato Sceces [] J Mordeso ad CSPeg uzzy Graphs d Hypergraphs Physca Verlag 000 [] agoor Ga ad M asheer hamed Order d Sze uzzy Graphs ullet of Pure ad ppled Sceces Vol E o p [3] agoor Ga ad S Shajtha egum Degree Order ad Sze tutostc uzzy Graphs teratoal Joural of lgorthms Computg ad Mathematcs33 00 [4] agoor Ga ad SR Latha O rregular uzzy Graphs ppled Mathematcal Sceces Vol6 o [5] SPatra Graph heory SKKatara & Sos ew Dehl [6] Rosefelduzzy Graphs uzzy Sets d her pplcatos o Cogtve ad Decso Process MEds cademc Press ew York [7] Shah Hussa eutrosophc Soft Graphs eutrosophc Sets ad Systems vol [8] Shah Some Studes eutrosophc Graphs eutrosophc Sets ad Systems Vol [9] Shah Regular eutrosophc Graphs Submtted [0] G S Sgh Graph heory PH Learg lmted ew Delh [] Smaradache eutrosophc Set a geeralsato of the tutostc fuzzy sets ter JPure ppl Math [] C Vasudev Graph theory wth applcatos 06 [3] L Zadeh uzzy Sets formato ad Cotrol [4] W-RZhag polar uzzy Sets d Relatos Computatoal ramework or Cogtve Modelg d Multaget Decso alyss Proceedgs of EEE Cof994pp Receved: ovember 0 06 ccepted: December 5 06 asr Shah ad Sad roum rregular eutrosophc Graphs

57 eutrosophc Sets ad Systems Vol eutrosophc eatures for mage Retreval Salama Mohamed Esa Hewayda ElGhawalby 3 Eawzy 4 Port Sad Uversty aculty of Scece Departmet of Mathematcs ad Computer Scece drsalama44@gmalcom 4 Port Sad Uversty Hgher sttute of Maagemet ad Computer Computer Scece Departmet mmmesa@yahoocom 3 Port Sad Uversty aculty of Egeerg Physcs ad Egeerg Mathematcs Departmet ayafawzy36@gmalcom bstract he goal of a mage Retreval System s to retreve mages that are relevat to the user's request from a large mage collecto ths paper we preset teture features for mages embedded the eutrosophc doma he am s to etract a set of features to represet the cotet of each mage the trag database to be used for the purpose of retrevg mages from the database smlar to the mage uder cosderato Keywords: Cotet-ased mage Retreval CR etbased mage Retreval R eutrosophc Doma eutrosophc Etropy eutrosophc Cotrast eutrosophc Eergy eutrosophc Homogeety troducto mage Retreval System s a computer system for searchg ad retrevg mages from a large database of dgtal mages he tradtoal way to mage retreval s the tet-based mage retreval R whch proposed late 970s [7 43] Such techques commece by aotatg the mages by tet ad the use tet-based database maagemet systems to retreve mages [8] lthough there are several progresses have bee made to R techques Such as data modelg Multdmesoal deg ad query evaluato there are some lmtatos whe usg such techques or stace the problem of aotatg mages large volumes of databases ad that oly oe laguage s vald for mage retreval urthermore the problems due to the subjectvty of huma percepto arsg from the resposblty of the ed-user; as well as the queres that caot be descrbed at all but tap to the vsual features of the mage [ 3 4 5] Later o durg the 990's aother way was veted to retreve mages whch s Cotet-based mage Retreval CR techque he ew techques came up to deal wth the rapd crease of usg dgtal mages databases o the teret Used for retrevg maagg ad avgatg large dgtal mages databases the CR techques de the mages by ther ow vsual cotet such as color ad teture stead of aotated the mage maually by tet-based key words [ ] he eutrosophc logc whch proposed by Samaradache [40] s a geeralzato of fuzzy sets whch troduced by Zada at 965 [45] he fudametal cocepts of eutrosophc set troduced by Samaradache [4 4] ad Salama etl [ ] mage Retreval echque Cotet-ased mage Retreval CR he Cotet-ased mage Retreval was used to depct the epermets of automatc retreval mages from a database that depeded o colors ad shapes fter that t used to retreve mages from a large collecto of database based o sytactcal mage features CR used some techques tools ad algorthms whch take from some felds such as statstcs patter recogto sgal processg ad computer vso CR the mages deed by the descrpto of the vsual cotet of the mages Most of the CR systems are cocered wth the appromate queres because t s am to fd the mages vsually smlar to the target mage he target of CR system s to duplcate the huma percepto of mage smlarty as possble as t ca eature Etracto s the basc of CR eatures may cota both tet-based features key words aotatos ad vsual features color teture shape faces he goal of feature etracto s to create hgh-level data pel Salama Mohamed Esa Hewayda ElGhawalb Eawzy eutrosophc eatures for mage Retreval

58 eutrosophc Sets ad Systems Vol values he vsual features ordered three levels: low level features prmtve mddle level features logcal ad hgh level features abstract ll recetly system were depeded o low level features color shape ut ow both md-level ad hgh-level mage represetatos are demad he effcecy of a CR system depeds o etracted features [6] he stages of the CR process are: - mage acqusto: to acqure a dgtal mage [9] mage database: t cossts of the collecto of umber of mages whch depeds o the user rage ad choce [9] - mage processg: t used to mprove the mage by creased the chaces for success of the other processes t frst the mage processed to etract the features that depct ts cotets hs process cotas flterg ormalzato segmetato ad object detfcato or eample the process of mage segmetato s used to dvde a mage to multple parts ad ts output s a set of sgfcat regos ad objects [9] 3- eature etracto: the features such as shape teture color are used to depct the cotet of the mage he features ca be characterzed as low-level ad hgh-level features he vsual formato ths step etracted from the mage ad saved as feature vector a features database he mage descrpto for each pel s foud the form of feature vector by usg the feature etracto hese feature vectors are used to make a compare betwee query wth other mages ad retreval [9] 4- Smlarty matchg: for each mage ts formato stored ts feature vectors for computato process ad these feature vectors are matched wth the feature vectors of query mage to helps smlarty measure hs step cotas the matchg of the above stated features to have that s vsually smlar wth the use of smlarty measure method called as Dstace method here are aother dstace methods such as Eucldea dstace Cty block dstace Caberra dstace [9] 5- Resultat retreval mages: ths process searched for the pror mataed formato to fd the matched mages from database ts output wll be the smlar mages wth the same or very closest features as that of the query mage [8] 6- User terface ad feedback whch cotrols the dsplay of the outcomes ther rakg ad the type of user teracto wth possblty of refg the search by some automatc or maual prefereces scheme [4] Color eatures for mage Retreval Color s wdely used low-level vsual features ad t s varat to mage sze ad oretato [9] Color Hstogram: CR oe of the most popular features s the color hstogram HSV color space whch used MPEG-7 descrptor t frst the mages coverted to the HSV color space ad uformly quatzg H S ad V compoets to 6 ad regos respectvely geerates the 64-bt color hstogram [44] Color momets: o form a 9-dmesoal feature vector the mea µ stadard devato σ ad skew g are etracted from the R G color spaces he best kow space color ad commoly used for vsualzato s the RG space color t ca be depct as a cube where the horzotal - as as red values creasg to the left y-as as blue creasg to the lower rght ad the vertcal z-as as gree creasg towards the top [] eture eature for mage Retreval the teture feature etracto usg the gray level co-occurrece matr for the query mage ad the frst mage the database to etract the teture feature vector [9] he co-occurrece matr represetato s a techque used to gve the testy values ad the dstrbuto of the testes he features whch selected for retrevg teture propertes are Eergy Etropy verse dfferece Momet of erta Mea Varace Skewess Dstrbuto uformty Local statoary ad Homogeety [5] 3 Shape eatures for mage Retreval he shape defed as the characterstc surface cofgurato of a object: a outle or cotour he object ca be dstgushed from ts surroudgs by ts outle [9] We ca dvde the shape represetatos to two categores: - oudary-based shape represetato: t uses oly the outer boudary of the shape t works by descrbg the cosdered rego by usg ts eteral characterstcs or eample the pels alog the object boudary [39] - Rego-based shape represetato: t uses the etre shape rego t works by descrbg the cosdered rego usg ts teral characterstcs or eample the pels whch the rego cotaed [39] 3 mages the eutrosophc Doma wth Hestacy degree he mage the eutrosophc doma s cosdered as a array of eutrosophc sgletos [5] Let U be a uverse of dscourse ad W s a set U whch composed of brght pels eutrosophc mages Salama Mohamed Esa Hewayda ElGhawalb Eawzy eutrosophc eatures for mage Retreval

59 eutrosophc Sets ad Systems Vol s characterzed by three sub sets ad whch ca be defed as s the degree of membershp s the degree of determacy ad s the degree of omembershp the mage a pel P s descrbed as P whch belogs to W by t s t% s true the brght pel % s the determate ad f% s false where t vares vares ad f vares the mage doma the pel pj s trasformed to Where belogs to whte set belogs to determate set ad belogs to o-whte set Whch ca be defed as [7]: p S j { j j j g j g j m g ma g m H 0 j H 0 j H 0 ma H 0 m j j H 0 j abs g j g j Where g j ca be defed as the local mea value of the pels of wdow sze ad ca be defed as the homogeety value of at j whch descrbed by the absolute value of dfferece betwee testy g j ad ts local mea value he secod trasformato for Where [5] 4 eture features eutrosophc doma 4 eutrosophc Etropy Shaos Etropy provdes a absolute lmt o the best possble average legth of lossless ecodg or compresso of a formato source Geerally you eed bts to represet a varable that ca take oe of values f s a power of f these values are equally probable the etropy s equal to the umber of bts equalty betwee umber of bts ad shaos holds oly whle all outcomes are equally probable f oe of the evets s more probable tha others observatos of that evet s less formatve Coversely rare evets provde more formato whe observed Sce observato of less probable evets occurs more rarely the et effect s that the etropy receved from o-uformly dstrbuted data s tha Etropy s zero whe oe outcome s certa Shao etropy quatfes all these cosderatos eactly quatfes all these cosderatos eactly whe a probablty dstrbuto of the source s kow Etropy oly takes to accout the probablty of observg a specfc evet so the formato whch ecapsulates s formato about the uderlyg probablty dstrbuto ot the meag of the evets themselves [37] Etropy s defed as []: Etropy j P j log j lthough the eutrosophc Set Etropy was defed oe dmeso preseted [0] we wll defe t two dmeso to be as follows: EtS E E E Et j log j j P P Et j log j j P P Et j log j j P P where P cotas the hstogram couts ecause we used the terval betwee 0 ad may have egatve values So we use the absolute of 4 eutrosophc Cotrast Cotrast s the dfferece lumace or color that makes a object dstgushable vsual percepto of the real world cotrast s determed by the dfferece the color ad brghtess of the object ad other objects wth the same feld of vew he huma vsual system s more sestve to cotrast tha absolute lumace he mamum cotrast of a mage s the cotrast rato or dyamc rage t s the measure of the testy cotrast betwee a pel ad ts eghbor over the whole mage t ca be defed as [38]: Cotrast j P j j We wll defe the eutrosophc set Cotrast to be as follows: CotS Cot Cot Cot Cot j P j j Cot j P j j Cot j P j j Salama Mohamed Esa Hewayda ElGhawalb Eawzy eutrosophc eatures for mage Retreval

60 eutrosophc Sets ad Systems Vol eutrosophc Eergy t s the sum of squared elemets Whch defed as [3]: Eergy j j P We wll defe the eutrosophc set Eergy to be as follows: ES E E E E P j j E P j j E P 9 j j 44 eutrosophc Homogeety Homogeety descrbe the propertes of a data set or several datasets Homogeety ca be studed to several degrees of complety or eample cosderatos of homoscedastcty eame how much the varablty of data-values chages throughout a dataset However questos of homogeety apply to all aspects of the statstcal dstrbutos cludg the locato parameter Homogeety relates to the valdty of the ofte coveet assumpto that the statstcal propertes of ay oe part of a overall dataset are the same as ay other part metaaalyss whch combes the data from several studes homogeety measures the dfferece or smlartes betwee the several studes hat s a value whch measures the closeess of the dstrbuto of elemets Whch defed as [0]: p j Homogeety j j We wll defe the eutrosophc set Homogeety to be as follows: HomoS Homo Homo Homo P j Homo j j P j Homo j j P j Homo j j Recetly the Eucldea dstace s calculated betwee the query mage ad the frst mage the database ad stored a array hs process s repeated for the remag mages the database followed by storg ther values respectvely he array s stored ow ascedg order ad dsplayed the frst 8 closest matches 5 Cocluso ad uture Work ths paper we troduced a survey of the et- ased mage Retreval R ad the Cotet-ased mage Retreval CR We also troduced the mage eutrosophc doma ad the teture feature eutrosophc doma the future we pla to troduce some smlarty measuremet whch may be used to determe the dstace betwee the mage uder cosderato ad each mage the database usg the features we troduced ths paper Hece the mages smlar to the mage uder cosderato ca be retreved 6 Refereces [] lbow S Salama & Esa M ew Cocepts of eutrosophc Sets teratoal Joural of Mathematcs ad Computer pplcatos Research JMCR [] Chag S ad u K S Relatoal Database System or mages Pctoral formato Systems the Seres Lecture otes Computer Scece May 005 [3] Chag S ad u K S Query-by Pctoral-Eample EEE ras O Software Egeerg SE [4] Chag S K Pctoral Data-ase Systems EEE Computer 4 3- ov 98 [5] Chag S K Ya C W Dmtroff D C ad rdt tellget mage Database System EEE ras Software Eg [6] Cha YK ad Che CY mage Retreval System ased o Color Complety ad Color Spatal eatures J of Systems ad Software [7] Cheg H D Guot Y Zhag Y ovel mage Segmetato pproach ased o eutrosophc Set d mproved uzzy C- Meas lgorthm World Scetfc Publshg Compay ew Math d atural Computato [8] Datta R Dhraj L J Wag J Z mage Retreval: dea flueces ad reds of the ew ge CM Computg Survey [9] Dash M Rawat R Sharma R Survey: Cotet ased mage Retreval ased o Color eture Shape & euro uzzy Mohd Dash et al t Joural of Egeerg Research ad pplcatos Sep-Oct 03 Salama Mohamed Esa Hewayda ElGhawalb Eawzy eutrosophc eatures for mage Retreval

61 eutrosophc Sets ad Systems Vol [0] Esa M ew pproach or Ehacg mage Retreval Usg eutrosophc Set teratoal Joural of Computer pplcatos Jue 04 [] Gudvada V ad Raghava J V Specal ssue o Cotet-ased mage Retreval Systems EEE Computer Magaze [] Gag Zh L J owu ad Wu Y Local Patters Costraed mage Hstograms or mage Retreval 5 EEE teratoal Joural Coferece o EEE [3] Hear D ad aker M P Computer Graphcs Eglewood Clffs J: Pretce Hall ch [4] Haafy Salama ad Mahfouz K Correlato of eutrosophc Data teratoal Refereed Joural of Egeerg ad Scece RJES [5] gle D hata Sh Cotet ased mage Retreval usg Combed eatures teratoal Joural of Computer pplcatos prl 0 [6] Ja R Guest E Specal ssue o Vsual formato Maagemet Comm CM Dec 997 [7] Kchag S ad Hsu mage formato System: Where Do We Go rom Here? EEE ras O Kowledge ad Data Egeerg [8] KeKre H Survey of CR echques ad Sematcs teratoal Joural of Egeerg Scece ad echology JES May 0 [9] Kog H mage Retreval usg oth Color d eture eatures Departmet of formato scece & echology Helogjag Proceedgs of the Eghth teratoal Coferece o Mache learg ad Cyberetcs aodg July 009 [0] Kujk M dvaced Computer Graphcs Hardware Sprger99 [] Lee Mueesawag P Gua L utomatc Relevace eedback for Dstrbuted Cotet-ased mage Retreval CG EEEorg LEX Chp sgal processor MC6875/D Motorola 996 [] arasmhalu D Specal Secto o Cotet-ased Retreval Multmeda Systems [3] Petlad ad Pcard R Specal ssue o Dgtal Lbrares EEE ras Patt Recog d Mach tell [4] Saea K Rchara V rved V Survey o Cotet ased mage Retreval usg DP VLC ad DCD Joural of Global Research Computer Scece September 0 [5] Salama Smaradache ad Esa M troducto to mage Processg va eutrosophc echques eutrosophc Sets ad Systems [6] Salama Esa M Elhafeez S Lotfy M M Revew of Recommeder systems lgorthms Utlzed socal etworks based e-learg Systems & eutrosophc System eutrosophc Sets ad Systems [7] Salama ad Elagamy H eutrosophc lters teratoal Joural of Computer Scece Egeerg ad formato echology Reseearch JCSER [8] Salama asc Structure of Some Classes of eutrosophc Crsp early Ope Sets & Possble pplcato to GS opology eutrosophc Sets ad Systems [9] Salama Smaradache lblow S he Characterstc ucto of a eutrosophc Set eutrosophc Sets ad Systems [30] Salama El-Ghareeb H Mae M Smaradache troducto to Develop Some Software programs for Dealg wth eutrosophc Sets eutrosophc Sets ad Systems [3] Salama bdelfattah M Esa M Dstaces Hestacy Degree ad leble Queryg va eutrosophc Sets teratoal of Computer pplcatos September 04 [3] Salama Esa M bdelmoghy M M eutrosophc Relatos Database teratoal Joural of formato Scece ad tellget System [33] Salama roum S Rughess of eutrosophc Sets Elr ppl Math [34] Salama El-Ghareeb H Mae M Lotfy M M Utlzg eutrosophc Set Socal etwork alyss e- Learg Systems teratoal Joural of formato Scece ad tellget Systems [35] Salama lagamy H eutrosophc lters teratoal Joural of computer Scece Egeerg ad formato echology Research [36] Schatz ad Che H uldg Large Scale Dgtal lbrares Computer [37] Shao C E Mathematcal heory of Commucato he ell System echcal Joural Jully October 948 [38] Sha M Uda D Computer Graphcs aha McGraw Hll publshg compay lmted [39] Sfuzzama M slam M R ad l M Z pplcato of Wavelet rasform ad ts dvatage Compared to ourer rasform Joural of Physcal Sceces Salama Mohamed Esa Hewayda ElGhawalb Eawzy eutrosophc eatures for mage Retreval

62 eutrosophc Sets ad Systems Vol [40] Smaradache eutrosophc set- geeralzato of the tutostc uzzy set Grautar Computg EEE teratoal Coferece 38-4 may 006 [4] Smaradache eutrosophy ad eutrosophc Logc rst teratoal Coferece o eutrosophy eutrosophc Logc Set Probablty ad Statstcs Uversty of ew Meco Gallup US [4] Smaradache Ufyg eld Logcs: eutrosophc Logc eutrosophy eutrosophc Set eutrosophc Probablty merca Research Press Rehoboth M [43] amura H ad Yokoya mage Database Systems: Survey Patter Recogoto [44] old P P Survey o Cotet-ased mage Retreval / browsg Systems Eplotg Sematc [45] Zadeh L uzzy sets formato ad Cotrol Receved: ovember ccepted: December 06 Salama Mohamed Esa Hewayda ElGhawalb Eawzy eutrosophc eatures for mage Retreval

63 eutrosophc Sets ad Systems Vol Uversty of ew Meco russ Desg Optmzato usg eutrosophc Optmzato echque Mrdula Sarkar Samr Dey ad apa Kumar Roy 3 Departmet of Mathematcs da sttute of Egeerg Scece ad echology ShbpurPO-otac Garde Howrah-703 West egal dae-mal:mrdulasarkar86@yahoocom Departmet of Mathematcs sasol Egeerg CollegeVvekaada Sara sasol West egal da E-mal:samr_besus@redffmalcom 3 Departmet of Mathematcs da sttute of Egeerg Scece ad echology ShbpurPO-otac Garde Howrah-703 West egal dae-mal:roy_t_k@yahooco bstract: ths paper we develop a eutrosophc optmzato SO approach for optmzg the desg of plae truss structure wth sgle objectve subject to a specfed set of costrats ths optmum desg formulato the objectve fuctos are the weght of the truss ad the deflecto of loaded jot; the desg varables are the crosssectos of the truss members; the costrats are the stresses members classcal truss optmzato eample s preseted to demostrate the effcecy of the eutrosophc optmzato approach he test problem cludes a two-bar plaar truss subjected to a sgle load codto hs sgleobjectve structural optmzato model s solved by fuzzy ad tutostc fuzzy optmzato approach as well as eutrosophc optmzato approach umercal eample s gve to llustrate our SO approach he result shows that the SO approach s very effcet fdg the best optmal solutos Keywords: eutrosophc Set Sgle Valued eutrosophc Set eutrosophc Optmzato o-lear Membershp ucto Structural Optmzato troducto the feld of cvl egeerg olear structural desg optmzatos are of great mportace So the descrpto of structural geometry ad mechacal propertes lke stffess are requred for a structural system However the system descrpto ad system puts may ot be eact due to huma errors or some uepected stuatos t ths jucture fuzzy set theory provdes a method whch deals wth ambguous stuatos lke vague parameters o-eact objectve ad costrat structural egeerg desg problems the put data ad parameters are ofte fuzzy/mprecse wth olear characterstcs that ecesstate the developmet of fuzzy optmum structural desg method uzzy set S theory has log bee troduced to hadle eact ad mprecse data by Zadeh [] Later o ellma ad Zadeh [4] used the fuzzy set theory to the decso makg problem he fuzzy set theory also foud applcato structural desg Several researchers lke Wag et al [8] frst appled α-cut method to structural desgs where the o-lear problems were solved wth varous desg levels α ad the a sequece of solutos were obtaed by settg dfferet level-cut value of α Rao [3] appled the same α-cut method to desg a four bar mechasm for fucto geeratg problem Structural optmzato wth fuzzy parameters was developed by Yeh et al [9] Xu [0] used two-phase method for fuzzy optmzato of structures Shh et al [5] used level-cut approach of the frst ad secod kd for structural desg optmzato problems wth fuzzy resources Shh et al [6] developed a alteratve α-level-cuts methods for optmum structural desg wth fuzzy resources Dey et al [] used geeralzed fuzzy umber cotet of a structural desg Dey et al used basc t-orm based fuzzy optmzato techque for optmzato of structure Dey et al [3] developed parameterzed t-orm based fuzzy optmzato method for optmum structural desg lso Dey etal [4] Optmzed shape desg of structural model wth mprecse coeffcet by parametrc geometrc programmg such eteso taassov [] troduced tutostc fuzzy set S whch s oe of the geeralzatos of fuzzy set theory ad s characterzed by a membershp fucto a o- membershp fucto ad a hestacy fucto fuzzy sets the degree of acceptace s oly cosdered but S s characterzed by a membershp fucto ad a o-membershp fucto so that the sum of both values s less tha oe trasportato model was solved by Jaa et al[5]usg mult-objectve tutostc fuzzy lear programmg Dey et al [] solved two bar truss o-lear problem by usg tutostc fuzzy optmzato problem Dey et al [6] used tutostc fuzzy Mrdula SarkarSamr Deyapa Kumar Roy russ Desg Optmzato usg eutrosophc Optmzato echque

64 eutrosophc Sets ad Systems Vol optmzato techque for mult objectve optmum structural desg tutostc fuzzy sets cosder both truth membershp ad falsty membershp tutostc fuzzy sets ca oly hadle complete formato ot the determate formato ad cosstet formato eutrosophc sets determacy s quatfed eplctly ad truth membershp determacy membershp ad falsty membershp are depedet eutrosophc theory was troduced by Smaradache [7] he motvato of the preset study s to gve computatoal algorthm for solvg mult-objectve structural problem by sgle valued eutrosophc optmzato approach eutrosophc optmzato techque s very rare applcato to structural optmzato We also am to study the mpact of truth membershp determacy membershp ad falsty membershp fucto such optmzato process he results are compared umercally both fuzzy optmzato techque tutostc fuzzy optmzato techque ad eutrosophc optmzato techque rom our umercal result t s clear that eutrosophc optmzato techque provdes better results tha fuzzy optmzato ad tutostc fuzzy optmzato Sgle-objectve structural model szg optmzato problemsthe am s to mmze sgle objectve fuctousually the weght of the structure uder certa behavoural costrats o costrat ad dsplacemet he desg varables are most frequetly chose to be dmesos of the cross sectoal areas of the members of the structures Due to fabrcatos lmtatos the desg varables are ot cotuous but dscrete for beloggess of cross-sectos to a certa set dscrete structural optmzato problem ca be formulated the followg form Mmze W 0 subject to m d R j j where W represets objectve fucto s the behavoural costrats m ad are the umber of costrats ad desg varables respectvely gve set d of dscrete value s epressed by R ad ths paper m objectve fucto s take as W l ad costrat are chose to be stress of structures as follows wth allowable tolerace 0 for m where ad l are weght of ut th volume ad legth of elemet respectvely m s the umber of structural elemet ad are the th stress allowable stress respectvely 0 3 Mathematcal prelmares 3 uzzy set Let X be a fed set fuzzy set set of X s a object havg the form fucto : 0 : X where the X defed the truth membershp of the elemet X to the set 3 tutostc fuzzy set Let a set X be fed tutostc fuzzy set or S X s a object of the form X X where : X 0 ad : X 0 defe the truth membershp ad falsty membershp respectvely for every elemet of X 0 33 eutrosophc set Let a set X be a space of pots objects ad X eutrosophc set X s defed by a truth membershp fucto a determacy-membershp fucto ad de- ad a falsty membershp fucto oted by X ad are real stadard or o-stadard subsets of ]0 [ hat s X : X ]0 [ ad : ]0 [ : X ]0 [ ad so here s o restrcto o the sum of 0 sup sup sup 3 34 Sgle valued eutrosophc set Let a set X be the uverse of dscourse sgle valued eutrosophc set over X s a object havg the X where form : X 0 : X 0 ad : for all X 35 Complemet of eutrosophc Set X wth Complemet of a sgle valued eutrosophc set s c ad s defed by deoted by c c c Mrdula SarkarSamr Deyapa Kumar Roy russ Desg Optmzato usg eutrosophc Optmzato echque

65 eutrosophc Sets ad Systems Vol Uo of eutrosophc sets he uo of two sgle valued eutrosophc sets ad s a sgle valued eutrosophc set C wrtte as C whose truth membershp determacymembershp ad falsty-membershp fuctos are gve by ma c ma c for all X m c 37 tersecto of eutrosophc sets he tersecto of two sgle valued eutrosophc sets ad s a sgle valued eutrosophc set C wrtte as C whose truth membershp determacymembershp ad falsty-membershp fuctos are gve by m c m c for all X ma c 4 Mathematcal aalyses 4 eutrosophc optmzato techque to solve mmzato type Sgle-Objectve Let a olear sgle-objectve optmzato problem be Mmze f Such that j g b j m 0 j Usually costrats goals are cosdered as fed quatty ut real lfe problem the costrat goal caot be always eact So we ca cosder the costrat goal for less tha type costrats at least b j ad t may possble to eted to b j 0 j b hs fact seems to take the costrat goal as a eutrosophc fuzzy set ad whch wll be more realstc descrptos tha others he the LP becomes SO problem wth eutrosophc resources whch ca be descrbed as follows Mmze f 3 Such that j g b j m 0 j o solve the SO 3 we are presetg a soluto procedure for sgle-objectve SO problem 3 as follows j j Step-: ollowg warer s approach solve the sgle objectve o-lear programmg problem wthout tolerace costrats e g b wth tolerace of acceptace costrats e 0 o-lear programmg techque Here they are Sub-problem- g b b by approprate j j j Mmze f 4 Such that j g b j m 0 Sub-problem- j Mmze f 5 Such that 0 g b b j m j j j 0 We may get optmal soluto * f f * * f f ad Step-: rom the result of step we ow fd the lower boud ad upper boud of objectve fuctos fu U U f f f be the upper bouds of truth determacy falsty fucto for the objectve respectvely ad L L L f f f be the lower boud of truth determacy falsty membershp fuctos of objectve respectvely the U U L L where 0 f f f f f f U f Lf U U L L where 0 f f f f f f U f Lf L L U L where 0 f f f f f f U f Lf Step-3: ths step we calculate membershp for truth determacy ad falsty membershp fucto of objectve as follows f f f f Lf U f f ep f L f U f f U f L f 0 f f U f Mrdula SarkarSamr Deyapa Kumar Roy russ Desg Optmzato usg eutrosophc Optmzato echque

66 eutrosophc Sets ad Systems Vol f f f f Lf U f f ep f L f U f f U f L f 0 f f U f f f 0 f f Lf U L f f tah f f f Lf f U f f f U f where are o-zero parameters prescrbed by the decso maker Step-4: ths step usg epoetal ad hyperbolc membershp fucto we calculate truth determacy ad falsty membershp fucto for costrats as follows gj g j f g j bj U g j j g 0 ep f bj g j bj bj Ug j L g j 0 0 f g j bj bj gj g j f g j bj bj g j j g ep f bj g j bj gj gj 0 f g j bj gj gj g j 0 f g j bj gj 0 bj bj g tah j 0 g j g f b j j g g j j bj bj 0 f g j bj bj where are o-zero parameters prescrbed by the dec- 0 so maker ad for j m 0 g g b j j j Step-5: ow usg SO for sgle objectve optmzato techque the optmzato problem ca be formulated as Mamze 6 Such that ; g j ; f ; ; g j f ; ; g j f 3; ; ; 0 where j m f g D for j m f g j j m f g D for j m ad f g j m f g for D j m j f g j are the truth determacy ad falsty membershp fuc- to of decso set D f g m j j ow f olear membershp be cosdered the above problem 6 ca be reduced to followg crsp lear programmg problem Mamze 7 Such that U L f f f U f U b f ; f U f f b g b b 0 j 0 j j j ; j f ; L ; f f f 0 g j g g b g j g j b b j ; 0 j j gj 3; ; ; ; Mrdula SarkarSamr Deyapa Kumar Roy russ Desg Optmzato usg eutrosophc Optmzato echque

67 eutrosophc Sets ad Systems Vol where gj b 0 j l ; 4; 6 j f U for j m l ; tah f 6 L f hs crsp olear programmg problem ca be solved by approprate mathematcal algorthm 5 Soluto of Sgle-objectve Structural Optmzato Problem SOSOP by eutrosophc Optmzato echque o solve the SOSOP step of 4 s used ad we wll get optmum solutos of two sub problem as ad fter that accordg to step we fd upper ad lower boud of membershp fucto of objectve fucto as UW UW UW ad U U U W W W where W U ma W W L m W W W U U L L where 0 U L W W W W W W W W L L U L where 0 U L W W W W W W W W Let the o-lear membershp fucto for objectve fucto W be W W W f W L U W ep f L W U W UW L W 0 f W UW W W W W W f W L U W ep f L W U W UW L W 0 f W UW W W ; f j g ep f 0 f 0 f 0 bj bj tah 0 f f 0 where are o-zero parameters prescrbed by the dec- 0 so maker ad for j m 0 the eutrosophc optmzato problem ca be formulated as Ma 8 such that ; W W W ; ; W W ; ; W 3; ; 0 he above problem ca be reduced to followg crsp lear programmg problem for o-lear membershp as Mamze 9 such that U W W W W W L W U U ; W U W ; L ; W W W W 0 0 ; Mrdula SarkarSamr Deyapa Kumar Roy russ Desg Optmzato usg eutrosophc Optmzato echque

68 eutrosophc Sets ad Systems Vol ; 0 3; ; ; 0 where l ; 4; W U ; W 6 L W l ; tah ad U ; 6 L hs crsp olear programmg problem ca be solved by approprate mathematcal algorthm 6 umercal llustrato well-kow two-bar [7] plaar truss structure s cosdered he desg objectve s to mmze weght of the structural W y of a statstcally loaded twobar plaar truss subjected to stress y costrats o each of the truss members ; Such that P l y y ; C l P y y ; C C l 05 y 5 0 0; where P odal load ; volume desty ; l legth of C ; perpedcular dstace from C to pot Cross secto of bar- ; Cross secto of bar- C mamum allowable tesle stress C mamum allowable compressve stress ad y y -co-ordate of ode put data for crsp model 0 s table Soluto : ccordg to step of 4we fd upper ad lower boud of membershp fucto of objectve fucto as U U U W W W ad L L L W W W where U L W W L 4393 U W L W W U L W W 0 W 6665 W W W gure Desg of the two-bar plaar truss he mult-objectve optmzato problem ca be stated as follows Mmze W y l y y 0 ow usg the bouds we calculate the membershp fuctos for objectve as follows W y W y f W y W y f W y 4393 ep 4 f W y 4393 W y W y f W y W W y ep f W y W 0 f W y W W Mrdula SarkarSamr Deyapa Kumar Roy russ Desg Optmzato usg eutrosophc Optmzato echque

69 eutrosophc Sets ad Systems Vol W y W y 0 f W y W W tah W y f W W y W f W y 4393 Smlarly the membershp fuctos for tesle stress are y y f y y 0 0 f y 50 ep 4 f 30 y 50 y y f y y ep f 30 y 30 0 f y 30 y y y y C C f C y y 0 0 f C y 00 C ep 4 f 90 C y 00 y C y 0 f y 30 ow usg above metoed truth determacy ad 80 6 falsty membershp fucto LP 7 ca be solved by tah y f y 0 SO techque for dfferet values of W ad C f y 50 W he optmum soluto of SOSOP0 s C where gve table ad the soluto s compared wth fuzzy 0 0 ad tutostc fuzzy optmzato techque ad the membershp fuctos for compressve stress costrat are able : put data for crsp model 0 C f C y C C y ep f 90 C y 90 C C 0 f C y 90 C C y C y 0 f C y 90 C 90 6 C tah C y f C C y 0 C f C y 00 where 0 0 C C ppled load P K Volume desty 3 K / m Legth l m Mamum allowable tesle stress Mpa Mamum allowable compressve stress C Mpa Dstace of from C m wth tolerace 0 90 wth tolerace 0 able : Comparso of Optmal soluto of SOSOP 0 based o dfferet methods Methods m m W K y m uzzy sgle-objectve o-lear programmg SOLP wth o-lear membershp fuctos tutostc fuzzy sgle-objectve olear programmg SOLP wth o-lear membershp fuctos eutosophc optmzatoso wth o-lear membershp fuctos Mrdula SarkarSamr Deyapa Kumar Roy russ Desg Optmzato usg eutrosophc Optmzato echque

70 eutrosophc Sets ad Systems Vol Here we get best solutos for the dfferet tolerace ad 3 for determacy epoetal membershp fucto of objectve fuctos for ths structural optmzato problem rom the table t shows that SO techque gves better Pareto optmal result the perspectve of Structural Optmzato 7 Coclusos he ma objectve of ths work s to llustrate how eutrosophc optmzato techque usg o-lear membershp fucto ca be utlzed to solve a olear structural problem he cocept of eutrosophc optmzato techque allows oe to defe a degree of truth membershp whch s ot a complemet of degree of falsty; but rather they are depedet wth degree of determacy he umercal llustrato shows the superorty of eutrosophc optmzato over fuzzy optmzato as well as tutostc fuzzy optmzato he results of ths study may lead to the developmet of effectve eutrosophc techque for solvg other model of olear programmg problem other egeerg feld Refereces [] taassov K tutostc fuzzy sets uzzy Sets ad Systemsvol0 ssue [] Zadeh L uzzy set formato ad Cotrol vol8 ssue [3] Rao S S Descrpto ad optmum desg of fuzzy mechacal systems Joural of Mechasms rasmssos ad utomato Desg vol09ssue [4] ellma R E &Zadeh L Decso-makg a fuzzy evromet Maagemet scece vol7 ssue [5] Shh C J & Lee H W Level-cut approaches of frst ad secod kd for uque soluto desg fuzzy egeerg optmzato problems amkag Joural of Scece ad Egeergvol7ssue [6] Shh C J Ch C C & Hsao J H lteratve α-levelcuts methods for optmum structural desg wth fuzzy resources Computers & structures vol8ssue [7] Smaradache eutrosophy eutrosophc probablty set ad logc mer Res Press Rehoboth US [8] Wag GY & Wag WQ uzzy optmum desg of structure Egeerg Optmzato vol [9] Yeh YC & Hsu DS Structural optmzato wth fuzzy parameters Computer ad Structurevol37ssue [0] Chagwe X uzzy optmzato of structures by the twophase method Computers & Structures vol3ssue [] Dey S & Roy K uzzy programmg echque for Solvg Mult-objectve Structural Problem teratoal Joural of Egeerg ad Maufacturgvol 4ssue [] Dey S & Roy K Optmzed soluto of two bar truss desg usg tutostc fuzzy optmzato techque teratoal Joural of formato Egeerg ad Electroc usess vol6ssue [3] Dey S & Roy K Mult-objectve structural desg problem optmzato usg parameterzed t-orm based fuzzy optmzato programmg echque Joural of tellget ad uzzy Systems vol30ssue [4] Dey S & Roy Optmum shape desg of structural model wth mprecse coeffcet by parametrc geometrc programmg Decso Scece Letters vol4ssue [5] Jaa & Roy K Mult-objectve tutostc fuzzy lear programmg ad ts applcato trasportato model otes o tutostc uzzy Sets vol3ssue [6] Dey S & Roy K Mult-objectve structural optmzato usg fuzzy ad tutostc fuzzy optmzato techque teratoal Joural of tellget systems ad applcatos vol7 ssue [7] l ehda K & awaz Z pplcablty ad vablty of a G based fte elemet aalyss archtecture for structural desg optmzato Computers & structures vol8ssue Receved: ovember 06 ccepted: December 8 06 Mrdula SarkarSamr Deyapa Kumar Roy russ Desg Optmzato usg eutrosophc Optmzato echque

71 eutrosophc Sets ad Systems Vol Uversty of ew Meco Las habldades del marketg como determates que sustetara la compettvdad de la dustra del arroz e el cató Yaguach plcacó de los úmeros SV a la prorzacó de estrategas Marketg sklls as determats that uderp the compettveess of the rce dustry Yaguach cato pplcato of SV umbers to the prortzato of strateges Pablo José Meédez Vera Crstha abá Meédez Delgado Mram Peña Gózalez 3 Makel Leyva Vázquez 4 Uversdad Espírtu Sato Samborodó Guayas Ecuador E-mal: pablomv_63@hotmalcom Uversdad Espírtu Sato Samborodó Guayas Ecuador E-mal: crstha-meedez@hotmalcom 3 Uversdad de Guayaqul acultad de Cecas Matemátcas y íscas Guayaqul Ecuador E-mal: mepg60@gmalcom 4 Uversdad de Guayaqul acultad de Cecas Matemátcas y íscas Guayaqul Ecuador E-mal: mleyvaz@gmalcom bstract: Ecuador specfcally the Yaguach Cato there s a eormous potetal the rce producto whch ufortuately s ot beg well used ad drve by marketg strateges ths work marketg strateges were developed that help to susta the commercal actvty of rce Yaguach Cato ad ts surroudgs he proposed strateges were aalyzed ad prortzed usg SV ad Eucldea dstace for the treatmet of eutraltes he paper eds wth cocluso ad future work proposal for the applcato of eutrosophy to ew areas of marketg Keywords: rce marketg eutrosophy SV umbers RODUCCO tecedetes: Cuado escuchamos la palabra Marketg uestro poscoameto cogosctvo os erumba a detallar mágees que e el mudo actual está presetes e toda gestó de oferta y demada os dejamos llevar por la fatasía de la publcdad y os dejamos evolver por los curosos y etretedos vdeos que recrea uestra decsó de compra uestra mete se ocupa cas e su totaldad e productos de áreas específcas de la sutuosdad dversó mercado del etretemeto etre otros; pero muy poco relacoamos productos de prmera ecesdad tales como arroz azúcar hara etc co estrategas de Marketg E Ecuador específcamete e el Cató Yaguach este u eorme potecal e la dustra de la Producco- Comercalzaco del arroz que desafortuadamete o está sedo be aprovechada y erumbada por estrategas de la Mezcla del Marketg que le de stal ecesaro para que esta dustra sga crecedo y cotrbuyedo a las ecoomías sostebles y al mpacto que ejerce e lo laboral ofrecedo empleo e todas las áreas que egloba las actvdades del agro-comerco e el país Es por esto que e este presete estudo se desarrollara estrategas de marketg que ayude a sosteer la actvdad comercal del arroz proveete de la zoa geográfca de Yaguach y sus alrededores plateádoos objetvos alcazables y justfcado el accoar de los modelos coceptuales que sostega las Pablo José Meédez Vera Crstha abá Meédez Delgado Mram Peña Gózalez Makel Leyva Vázquez Marketg sklls as determats that uderp the compettveess of the rce dustry Yaguach cato pplcato of SV umbers to the prortzato of strateges

72 eutrosophc Sets ad Systems Vol estrategas a desarrollarse cocluyedo co ua propuesta erquecedora e toro a la ueva forma de maejar la compettvdad dferecádoos e las habldades s descudar las debldades que os ayudara a erquecer la estratega de crecmeto y compettvdad e el mercado de cosumo recomedado stuacoes ecotradas o alcazables medbles e el presete estudo Defcó del Problema: El arroz es el cultvo que más etesó abarca e el Ecuador ocupado Has segú la Ecuesta de Superfce y Produccó gropecuara Cotua ESPC 05 [] La mayor área sembrada de arroz e el país se ecuetra e la regó Costa pero també se sembra e las estrbacoes adas y e la mazoía pero e catdades poco sgfcates abla REGÓ Superfce Has Sembro Cosecha Producco m Vetas m otal acoal Rego Serra Rego Costa Rego Orete Zoa o Delmtad abla Superfce producco y vetas segu rego uete: EC 05 Dos provcas Guayas y Los Ríos represeta el 94% de la superfce sembrada de la gramíea e el Ecuador E cuato a la produccó de forma correspodete Guayas y Los Ríos tee el 783% y % y 38% respectvamete abla PROV C Cañar SUPERCE Has Sembro Cosecha Producc m Veta m Loja Sato Domgo El Oro Esmeralda Guayas Los Ros Maab Moroa Orellaa Sucumbo abla # Segu Provcas uete: EC 05 La dustra del arroz tee su cocetracó máma etre tres provcas cas eclusvamete Guayas Los Ríos y Maabí que so quees marca las pautas de la sembra cosecha plada y comercalzacó [] E la dustra almetara la gramíea del arroz costtuye ua de las fuetes prcpales de la caasta famlar Por su parte el cató Yaguach fg cuya cabecera catoal se ecuetra a 9 km de Guayaqul preseta ua poblacó de habtates e u área de 5 km² Está asetada a 5 msm y su temperatura promedo es de 5 C su precptacó promedo aual está etre 500 y 000 mm [3] Su área cultvada de arroz correspode a 55 has [4] E otras stacas el problema se cocetra mucho mas e la comercalzaco porque las culturas y costumbres que los oblga a usar metodos caducos y pocos erumbados haca las tecologas o ayuda a mejorar las ofertas e u mercado e dode hay tradco de padres a hjos Las pladoras que es el termedaro que se desevuelve e el egoco del plado alguos au sgue hacedolo de maera muy artesaal regar el arroz e u tededero de secado al sol so muy poco los mas proactvos que ya ha empezado a tecfcar sus procesos e ua cadea de stalacoes dustralzados desde el tamzado hasta el secado e horos Los arrozeros se queja de que el gobero o apoya a las asocacoes yectado dero e las Pablo José Meédez Vera Crstha abá Meédez Delgado Mram Peña Gózalez Makel Leyva Vázquez Marketg sklls as determats that uderp the compettveess of the rce dustry Yaguach cato pplcato of SV umbers to the prortzato of strateges

73 eutrosophc Sets ad Systems Vol fraestructuras o tee agua o tee semllas de buea caldad el kt de sectcdas que oferta esta sedo mal dstrbudos pues solo se aprovecha los agrcultores cercaos El programa de alto redmeto estaba costtudo por u kt que traa 0 sumos: sacos de urea semllas sectcdas etre otros gura Cató Yaguach Platear estrategas de recoleccó de Datos de tpo epermetal y documetal caso empresaral para obteer resultados proyectados Propoer las estrategas de marketg para el crecmeto sosteble de la dustra del arroz e el cató Yaguach Prorzar las dsttas estrategas utlzado lógca eutrosofca 3 Justfcacó del Problema: Debdo a este feómeo el Cató Yaguach efreta codcoes desfavorables clusve e la provca del Guayas para mateerse actva comercalmete sguedo patroes acestrales de procesos e la produccó y específcamete por el descoocmeto de estrategas de marketg que motve al cosumo de la gramíea pues co el crecmeto desmeddo que preseta la dustra del tess e las socedades moderas e dode el patró de la belleza colleva dejar de cosumr productos de altos porcetajes de carbohdratos el arroz es el prmer producto que se evta cosumr e las detas almetcas y cosderado aú más que el arroz peruao está sedo cocebdo como ua gramíea co mejor caldad y meor preco e el mercado de cosumo ahodado más la stuacó de los productores del sector de Yaguach y sus alrededores El problema de este estudo se cocetra etoces e la falta de estrategas de mercadoteca que ayudaría a crecer a las empresas que coforma la dustra de produccó y comercalzacó del arroz e el cató Yaguach 4 Objetvos: Objetvo Geeral El objetvo fudametal de este proyecto es Propoer estrategas de comercalzacó sustetadas e el m del marketg que promueva y ayude al crecmeto sosteble de las empresas de la dustra del arroz e el cató Yaguach Objetvos Específcos Descrbr las estrategas de Marketg hacedo u aálss macro y mcro ambetal 5 Justfcacó de la vestgacó: S cosderamos que el problema esta crcuscrto e u producto de cosumo masvo y que la gramea es u susteto socal e las localdades mas olvdadas e el avace tecologco y de crecmeto estructural el cato Yaguach vee a cotrbur al sostemeto almetco del Ecuador el msmo que debe ser erumbado e ua poltca socal de cluso e los programas guberametales Es por esto que este estudo cocetro la metodologa del aalss de la voz drecta del agrcultor plador; para este proposto elegmos ua famla muy represetatva e la sembra y plado del arroz quees ademas ha cado co procesos artesaales llegado a los actuales mometos a poseer procesos dustrales be estructurados y co plaes de crecmeto sosteble Es el caso de la empresa mecorpoc S la msma que fue fudada el año 009 despues de haberse separado del ucleo famlar cal co quees tambe llego a sem dutralzar el proceso o cocluyedolo E la actualdad esta empresa se dedca a la sembra plada compra veta y comercalzaco del arroz esta ubcada e el km 5 de la va Yaguach-Juja La msma que os dara los parametros dspesables para el aalss estrategco 6 Marco eórco de la dustra del arroz e el Ecuador El eje de las estadístcas referecales os ha permtdo crear el coteto del problema y s os efocamos haca al aspecto utrcoal el arroz es el almeto más cosumdo e el mudo y ua quta parte de la poblacó del plaeta depede de su cultvo recetes estudos asoca u mayor cosumo del arroz co el aumeto de la obesdad y de factores de resgo de padecer efermedades cardovasculares y dabetes tpo [5 6] S embargo este práctcas de cultvos varedades y formas de cosumrlos que mtga o elma tato los mpactos medoambetales como los utrcoales Esta vsó os hace cosderar ua ameaza la falta de estrategas de marketg e las accoes comercales de la gramíea Por otro lado este varos factores que está afectado el cultvo y produccó de arroz a vel mudal E cuato al aspecto medoambetal su produccó esta asocada de etre el 7-7% de todas las emsoes de metao de orge Pablo José Meédez Vera Crstha abá Meédez Delgado Mram Peña Gózalez Makel Leyva Vázquez Marketg sklls as determats that uderp the compettveess of the rce dustry Yaguach cato pplcato of SV umbers to the prortzato of strateges

74 eutrosophc Sets ad Systems Vol humao y sedo el metao u potete gas de efecto veradero represeta u aporte sgfcatvo al caletameto global [7] Otro aspecto egatvo es el empleo de grades volume de agua para aegar los terreos[5] Co estos datos solo referecamos alguas de las accoes del marco teórco que os hará recorrer los coceptos de grades vestgadores y pesadores estratégcos[8] Comprometdos co el medo ambete y la ecooma sosteble el marketg be erubado y sustetado e la herrameta mas efcaz de aalss de datos OD os ayuda a vsualzar y delear mejor las estrategas para la comercalzaco de la gramea de los agrcultores de Yaguach E el sguete cuadro resummos lo epresado: ortalezas Oportudades Debldades meazas Producto cosumo masvo Descoocmet o de estrategas de mercadeo cluso socal e la Matrz Productva cultvos s cotrol del medo ambete las realdades del etoro os ayuda a efocar uestro aalss basadoos e las estrategas de PEERCO e el mercado sabedo que la compettvdad depedera de la fuerza compettva del sector y que la determara la retabldad del msmo MEODOLOG DE L VESGCO Se realza bajo u efoque cualtatvo que utlza recoleccó de datos e vestgacó prelmar [9 0] La msma se realzo e fuetes prmaras co etrevstas a las empresas de los hermaos Meedez Dseño de la vestgacó La costruccó del coteto cualtatvo so armadas desde la logca del aalss de caso y modelo y veles epstemológcos e la géess e hstora de la vestgacó socal[9] po de Estudo ee u alcace descrptvo y o epermetal porque se cosdera las característcas de la empresa co éto e el mercado del arroz y que tee poscoada la marca de RROZ MRVLL muy recoocda etre los compradores por su caldad El horzote espacal esta defdo por la actvdad del cultvo que actualmete está e esta dustra e el cató Yaguach y que so especfcamete agrcultores que terma su cclo co el plado del arroz acco que ya o depede de ellos so de ua operaco aslada sedo muy pocos los agrcultores que terma su proceso Despues de las reuoes establecdas e el sto se escogo ua empresa tpo para hacer el estudo de la msma se resume lo sguete: La empresa que os srve de aalss de caso es ecuatoraa deomada mecorpoc S esta ubcada e el km 5 de la va Yaguach-Juja fue fudada el año 009 por Jose vlo Meedez Medoza e u acto de maculacó los fodos utlzados para la creacó de mecorpoc S fuero recaudados medate apalacameto bacaro y ahorros producdos del trabajo efcaz y ehaustvo de vlo Meédez cuado era la cabeza prcpal de la pladora Hermaos Meédez la cual está ubcada a 6 cuadras del cetro del cató Yaguach Cuado sedo su admstrador logro dejarla sem dutralzada la msma que o ha tedo gua evoluco crecmeto e los ultmos 7 años corooborado as que s admstraco ordeada y sstematca o se cosgue resultados optmos mecorpoc perteece al sector gropecuaro y su prcpal actvdad es la produccó y comercalzacó del arroz E la producco esta empresa se dedca a la sembra y plada cueta co ua haceda co u área de 68 Hectareas Co ua produccó de 70 sacas de 00 lbs por hectárea tee cclos de sembra la de vero e el mes de eero y se cosecha e el mes de abrl 4 meses de cultvo y la de verao se sembra e julo y se cosecha e ovembre El costo de produccó por hectárea es de $400 el costo de u saco de las 00 lbs de arroz es de $4 E los meses de mayo y juo se prepara la terra para su respectva sembra al gual que el mes de dcembre La varedad de arroz que se sembra es el 09 f 5 Que so arroces grao largo de 4 meses cabe recalcar que hay arroces de cclo corto y a su vez se lo cooce como ap 4 o grao corto co ese tpo de arroz o trabaja por cuato solo tee mercado para arroces grao largo E[ arroz ua vez cosechado se lo lleva a la pladora para su debdo pílado el costo de cosecha y traslado a la plata es de $330 E la plata se lo pesa por báscula maejada por sstemas y se lo deposta e los slos de almaceameto este 3 slos de 3000 qutales cada uo para que después vaya a las dos secadoras de flujo cotuo para su respectvo secado e càscara la secadora puede secar 600 qutales cada horas Luego de este proceso de secado se lo almacea por 7 días para pasar a su respectvo pílado La píladora es de marca SKE Japoesa tee ua capacdad de plado de 0qq por hora y ua Pablo José Meédez Vera Crstha abá Meédez Delgado Mram Peña Gózalez Makel Leyva Vázquez Marketg sklls as determats that uderp the compettveess of the rce dustry Yaguach cato pplcato of SV umbers to the prortzato of strateges

75 eutrosophc Sets ad Systems Vol capacdad dara por 0 horas arrojado ua producco de 8800 qutales mesuales E la Comercalzaco se dedca a la Compra-veta compra arroz atural a los campesos del medo y lo almacea para el evejecdo atural lo que colleva a mejorar el preco cuado hay sobreproducco; o les da servco de plado y secado Etre los productos que maeja so: plado atural evejecdo atural segu el tempo y tee dervados tales como: ½ arrocllo ¾ arrocllo tza yele y polvllo Detro del arroz plado atural la empresa comercalza dferetes varedades como 09 f- f-5 y correte Posee dos marcas comercales sedo estas: GR RROZ MRVLL y RROZ COEJO La presetacó de sus sacos predoma los colores: araja café claro y egro e el saco Maravlla se evasa el arroz evejecdo aturalmete metras que e el saco coejo se evasa el arroz plado atural los demás dervados del arroz so evasados e sacos blacos s dseños Uo de los prcpales problemas para uestro arroz acoal es el greso del arroz peruao por cotrabado ya que este tpo de arroz tee mejores característcas que el arroz Ecuatorao esto se le atrbuye al tpo de clma que tee Perú u clma de mayor lumosdad óptmo para el proceso de fotosítess para la plata e comparacó a uestro clma Por las característcas del arroz Peruao u arroz mas brlloso de meos almdó de meor costo y de mejor preseca los cletes prefere rse por ese tpo de arroz lo cual a mermado las vetas 3 strumeto de la vestgacó Usamos el Método Delph que os permtó elaborar escearos futuros hacedo coverger la opó de u grupo de epertos que os ayudaro hacer estudos de tedecas y verfcacoes co la correlacó de los cosumdores frete a dversdad de temas que tee que ver co la dustra del cosumo de la gramíea 4 álss e terpretacó de resultados Ua vez realzados el aálss del caso empresaral se terpreta los resultados desde el puto de vsta de las valoracoes segú los étos y fracasos 3 MRCO EORCO Este dferetes tpos de estrategas e este trabajo presetaremos ua varedad agrupada de la sguete maera: Estrategas tesvas Crecmeto sosteble Estrategas de Dversfcaco abrr haca uevos productos Estrategas Defesvas vetaja compettva e el sector g : pos de Estrategas[] 3 Estrategas de Marketg adoptadas e el Crecmeto Sosteble Las teorías que se aalza e el presete proyecto está sustetadas e la lógca de la mercadoteca co la cual la udad de egocos espera alcazar sus objetvos y cosste e defr estrategas específcas para mercados meta cosderado hacer u redseño profudo del egoco para determar los plaes estrategcos [] El marketg puede producrse e cualquer mometo e que ua persoa o ua orgazacó se afae por tercambar algo de valor co otra persoa u orgazacó e este coteto el marketg costa de actvdades deadas para geerar y facltar tercambos co la tecó de facltar ecesdades o deseos de las persoas o las orgazacoes [3] Segu el modelo del aalss de las 5 fuerzas de Mchel Porter las fuerzas compettvas que formula la estructura basca de u sector y defe su retabldad so: Competdores actuales y su tesa rvaldad Etrada de uevos competdores Preso de productos susttutvos Epereca egocadora de los cletes Epereca egocadora de los proveedores l o costturse e u sector de fuerte crecmeto o se debe desarrollar estrategas muy agresvas ya que o se gaara cuota de mercado pero s se permtra corregr errores Los errores a corregrse saldra del aalss de sus practcas de procedmetos e la oferta y demada Debe mplemetarse cambo tecologco para rejuveecer el sector y aprovechar el apoyo guberametal e poltcas stauradas e la Matrz Productva del pla acoal co u acercameto de propuestas regularzadas por plaes de egocos formales que sustete el RO La epereca coqustada por el coocmeto del maejo del producto debe ser rescatada e u Pablo José Meédez Vera Crstha abá Meédez Delgado Mram Peña Gózalez Makel Leyva Vázquez Marketg sklls as determats that uderp the compettveess of the rce dustry Yaguach cato pplcato of SV umbers to the prortzato of strateges

76 eutrosophc Sets ad Systems Vol pla de mercadeo eaustvo del maejo de la promoco y publcdad tomado e cosderaco los dversos segmetos as se tedra e cueta al cosumdor morsta tato como al cosumdor mayorsta Debe prorzarse el trabajo de los proveedores ya que de esto depede e gra medda el crecmeto sosteble pues s las catdades adecuadas de la producco de lagramea se muere la lea del crecmeto y mucho mas el sostemeto de la dustra e la rego 3 Estrategas de Marketg adoptada e la Dversfcaco El cclo de vda del producto determa la madurez del sector y e uestro caso especfco del arroz afecta profudamete a la Gesto omado el aalss del cclo de vda del producto e todas sus etapas teemos: Desde la aparcó del marketg muchas corretes deológcas ha surgdo dado lugar a Modelos que puede ser usados o adaptados a dferetes etoros o crcustacas depededo del producto o servco que se va a propoer[5] Hay tres tpos de estrategas de Dversfcaco: Cocetrca: adcoar productos uevos pero relacoados para uevos cletes Horzotal: adcoar productos uevos s relacoarse y para cletes actuales Coglomerada: es la suma de productos uevos o relacoados aqu juega papel mportate la ovaco uestra decso es la Dversfcaco Cocetrca ya que os determa aprovechar los subproductos tales como la cascarlla que se usa como combustble y bo combustble y el desperdco tales como arrozllo grueso y fo los msmos que se coverte e polvllo covrtedolos e productos de uevos cletes que se cocetra e elaborar balaceados Debe ser fabrcados bajo ua ueva marca que permta epadr el mercado y afazar la estratega compettva g cclo de vda del sector [4] ases cclo competeca Vetas Estratega Producto troducco Etra pocos Poco Uco eefcos egatvos crecmeto Etra umeta Mejorarlo muchos eefcos mplar la postvos marca Crear marca madurez Gra competeca Vetas mamas eefcos altos declve Dsmuye Vets y beefcos dsmuye g 3 ases de aalss del cclo de vda Dferecarlo uevos usos Segmetos uevos Modfcarlo Elmarlo sustturlo 33 Estrategas de Marketg adoptada e la Vetaja Compettva U mercado de competeca perfecta es aquel que carece de barreras de etrada y salda y cuyos productos está estadarzados es decr cualquera puede etrar y competr e el egoco y los compradores adquere los productos eclusvamete e fucó del preco[6] E u mercado co esas caracterstcas o sera posble obteer beefcos a medao y a largo plazos embargo debdo a sus caracterstcas estructurales los mercados preseta alguas mperfeccoes marcas patetesregulacoes guberametales Debera trabajarse u pla de Marca el msmo que debe relacoarse co los colores y el ombre que ya esta poscoado como es el de RROZ MRVLL Pablo José Meédez Vera Crstha abá Meédez Delgado Mram Peña Gózalez Makel Leyva Vázquez Marketg sklls as determats that uderp the compettveess of the rce dustry Yaguach cato pplcato of SV umbers to the prortzato of strateges