An Application of Fuzzy Soft Set in Multicriteria Decision Making Problem

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1 Interntion Journ of Comuter Aition ( ) Voume 8 No, Jnury 0 An Aition of Fuzzy Soft Set in Mutiriteri Deiion Mking Probem P K D Dertment of Mthemti NERIST Nirjui, Arunh Preh R Borgohin Dertment of Mthemti NERIST Nirjui, ArunhPreh ABSTRACT The eiion mking robem with imreie t h ei ignifine in re ife robem Here the onet of fuzzy oft et whih wy oe rmeteriztion too i ie to ove muti-oberver muti-riteri eiion mking robem Keywor Soft et, Fuzzy oft et, Mrket Reerh Grou (MRG), Comrion tbe INTRODUCTION Mny of the re ife robem in engineering, mei iene, environment n oi iene, mngement et often invove t whih re not reie n eterminiti in hrter Thi i beue uh robem re eentiy humniti n more ubjetive in nture n o, they nee to be hne ifferenty thn the one with reie mthemti t Some of the reent theorie eveoe for hning robem with imreie t re robbiity theory, interv mthemti, fuzzy et,, rough et et But Mootov[] h hown tht eh of the bove toi uffer from ome inherent imittion tht they k the rmeteriztion too n introue Soft et theory hving rmeteriztion too for eing with vriou unertintie uefuy Subequenty Mji et [, ] extene oft et theory of Mootov n introue fuzzy oft et FurtherChuhuriet []hve few ition of fuzzy oft et with the he of the metho in [,] n omre them with robbiity itribution Ao Ҫğmn [5]ue the ff ggregtion oertor to form the eiion mking metho foowe by e tuy In 0 Neoget [6] ue fuzzy oft mtrie, fuzzy oft omement n fuzzy mtrix oertion to ove eiion mking robem Here we y fuzzy oft et in muti-oberver muti-riteri eiion mking robem n imrovement of the metho in [] SOFT SETS, FUZZY SOFT SETS AND THEIR OPERATIONS Definition Let be univere of ioure, E et of rmeter n A E Then i e oft et over, where F i ming given by F : A, the ower et of Equiventy, oft et over i rmeterize fmiy of ubet over the univere, ie, for e A, F( rereent the et of e-roximte member of the oft et, Exme Let {,, } be et of three rout n E { e (oty), e( vibii ty), e( ne} be the et of rmeter n A { e, e} E Then { F( e ) {, }, F( {, }} i oft et over Definition Let n ( G, be two oft et over ommon univere where A, B E Then (i) i ub oft et of ( G,, written ( if A B n F ( G(, e A (ii), ( G, if ( n ( G, (iii) The omement of oft et,, enote by, ( F, A ), where F : A uh tht F ( ~ F(, e A (iv) A oft et i i to be nu oft et, if e A, F(, where i the nu et of (v) AND oertion of two oft et, AND( enote by( H, A B ) (, i efine H(, ) F( ) G( ), (, ) A B (vi) OR oertion of two oft et, OR( enote by ( O, A B ) (, i efine O(, ) F( ) G( ), (, ) A B (vii) Interetion of two oft et n ( over the ommon univere i the oft et ( I, C) (, wherec A B n I : C uh tht I( F( or G(, e C

2 Interntion Journ of Comuter Aition ( ) Voume 8 No, Jnury 0 (viii)union of two oft et n ( over the ommon univere i the oft et ( U, C) (, wherec A B n U : C uh tht e C, F(, U( G(, F( G(, e A ~ B e B ~ A e A B Mthemti moeing of the robem Suoe there re m imir rout P {,,, } m in the mrket n the MRG hve tken ome eetion riteri S {,,, } for referene evution of the imir rout Their erformne evution i exree fuzzy oft et S) over P, where F : S P I, for eh MRG Definition Let be univer et, E et of rmeter n A E Ao et I enote the et of fuzzy ubet of Then ir i e fuzzy oft et over, where F i ming from A to I The efinition of ub fuzzy oft et, nu fuzzy oft et, interetion n union oertion re imir to thoe efine for ri oft et (oft et) Exme Conier,E, A given in Exme Then { F( e ) {(,0),(,05)}, F( {(,0),(,07)}} i fuzzy oft et over Now we tte ome bi reut on oft et/ fuzzy oft et 5 Prooition If n ( re two oft et (or fuzzy oft et then (i) Thu m m m m from MRG ( o ) R m (ii) (iii), (iv), (v) ( ( ) ( (vi) ( ( ) ( METHOD In thi etion, we reent eetion metho to buy the bet oibe rout from mong the imir rout tking into oniertion of the buyer referene A firt te, the buyer h to urvey the mrket to get n over knowege of the imir rout But if the rout i very exenive n the urhe i for ong time, then the buyer h to oet the erformne evution of the imir rout evute by ifferent reerh grou oerting in the mrket Suoe there re n MRG (mrket reerh grou), y { o, o,, on } roviing uh informtion on the bi of ertin eetion riteri ike ot, urbiity, mintenne, omfort et of the imir rout R m m m m m from MRG ( o )n o on Then tking the verge of the bove fuzzy oft et we get the erformne evution mtrix (or reent mrket urvey informtion)

3 Interntion Journ of Comuter Aition ( ) Voume 8 No, Jnury 0 R m m m where n ( k) ij ij k m ( ) m Suoe n iniviu, Mr, i interete to buy rout on the bi of the reent mrket informtion () But he my hve hi own weightge to ifferent eetion riteri ike, referene to urbiity, mintenne n ot tken in tht orer For exme referene weightge n be exree W w w w w Suh tht w Thi retrition i to mintin the fuzzy roerty of the memberhi vue Now to get the omrehenive eiion mtrix D for, we mutiy R T (trnoe of R) by the referene weightge mtrix uh tht D ( ij w j ) m Thu we get the omrehenive eiion mtrix D m m m m m Now we ontrut the omrion tbe for the rout to eie the bet oibe rout for Mr tht he my buy The omrion tbe i qure tbe with equ number of row n oumn where both row n oumn re bee by the rout nme,,, m n the entrie re ij, with i, j,,,, m given by the number of eetion riteri for whih the memberhi vue of exee or equ to the memberhi vue of ij j i Row um, oumn um n ore The row-um of rout i, enote by ri, n i ute by the formu r m i ij j Cery, ri inite the tot number of rmeter in whih i ominte the member of P Simiry the oumn-um of rout j, enote by, i ute by the formu Here o the integer rmeter in whih m j ij i j inite the tot number of j i ominte by the member of P The ore of rout i i i n i given i ri - j The equene of i ut in ereing orer give the orer of referene of the rout for the buyer Then the rout with mximum ore i the bet oibe otion for Mr In e we nee to eie the bet oibe rout for mutie buyer on the bi of the iniviu referene weightge then we tke ifferent omrehenive mtrix D for ifferent buyer reeting the ret of the ution ALGORITHM The foowing gorithm i uggete for the oution of the robem iue bove Inut the erformne evution of the imir rout by ifferent mrket reerh grou mtrie Fin the verge of the orreoning entrie of the mtrie in te I Mutiy the weightge of the eetion riteri of the utomer to the orreoning entrie of eh row to get the omrehenive eiion mtrix Formute the omrion tbe 5 Fin the row-um n oumn-um of the omrion tbe 6 Obtin the ore for eh rout n the rout with mximum ore i reommene the bet hoie 5 CASE STUDY The Suoe Mr i interete to buy r from mong the et of r C{,, }on the bi of the et S { (oty), ( omfort), ( fueeffiieny), ( mintinen} of eetion riteri e the rmeter n uoe Mr i interete to buy the r on hi own referene weightge to the eetion riteri Now to get the reent mrket informtion, ie, the erformne evution j 5

4 Interntion Journ of Comuter Aition ( ) Voume 8 No, Jnury 0 mtrix we ontrut the fuzzy oft et,, over C, where F, F, F re ming from S into I given by three MRG foow Suoe F ( ) { /8, /7, /}, F ( ) { /, /, /5}, F ( ) { /6, /, /} n F ( ) { /, /6, /7} Then the fuzzy oft et i rmeterize fmiy of fuzzy et over C n give oetion of roximte erition of the reent mrket informtion by the MRG Simiry uoe the oft et n,where F ( ) { /5, /8, /}, F ( ) { /7, /, /}, F ( ) { /9, /, /6} n F ) { /, /6, /8} ( F ( ) { /, F ( ) { /5, /7, /}, F ( ) { /, /9, /}, /7, /8} ( n F ) { / 6, /, / 8} wi give the roximte erition of the reent mrket informtion by MRG n MRG reetivey Now the mtrix rereenttion of the bove three fuzzy oft et,( F n re 8 7 5, n Then tking the verge of the bove three fuzzy oft et we get the erformne evution mtrix (or reent mrket urvey informtion) R Next, uoe tht the referene weightge of Mr to the ifferent eetion riteri i given by the mtrix W Thu to get the omrehenive eiion mtrix D for Mr, we mutiy R T by the referene weightge mtrix n get D foow: D The omrion tbe of the bove omrehenive eiion mtrix i: Next we omute the row-um, oumn-um from the omrehenive eiion mtrix n the ore for eh i beow: row-um oumn-um ore Cery the mximum ore i, ore by the r Therefore, the r i the bet hoie for Mr 6

5 Interntion Journ of Comuter Aition ( ) Voume 8 No, Jnury 0 6 CONCLUSION Mootov[] introue the oft et theory with rmeteriztion roerty for eing with unertin, imreie or fuzzy onet An ition of fuzzy oft theory i reente here in muti-riteri eiion mking robem uing muti-oberver erformne evution ong with the erformne weightge of the iniviu eiion mker It o invove ontrution of omrion tbe from the omrehenive eiion mtrix, rereenting oft et Further we oberve tht the reent metho n be ombine with tht in [5] 7 REFERENCES [] Mootov, D Soft et theory- Firt reut, Comut Mth A 7 (999), 9- [] Mji, PK, Biw, R, n Roy, AR Fuzzy oft et, J Fuzzy Mth 9 (), (00), []Mji, PK, Biw, R, n Roy, AR Soft et theory, Comt Mth A 5 (00), [] Chuhuri, A, De, K, n Chtterjee, D Soution of the Deiion mking robem uing fuzzy oft retion, Int J of Informtion Tehnoogy, 5(), 009, [5] Ҫğmn, N, Ҫitk, F, n Enginoğu, S Fuzzy rmeterize fuzzy oft et theory n it ition, Turkih J Fuzzy Sytem, (), 00, -5 [6] Neog, T J, n Sut, D K Aition of fuzzy oft et in eiion mking robem uing fuzzy oft mtrie, Int J of Mthemti Arhive, (), 0, 58-6 [7] Kir, G J, Yun, Bo 000 Fuzzy Set n Fuzzy Logi: Theory n Aition PHI 7