A FUZZY MULTI-ATTRIBUTE DECISION MAKING ALGORITM BASED ON INTUITIONISTIC FUZZY SETS

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1 FUZZY MULTI-TTRIBUTE DECISION MKING LGORITM BSED ON INTUITIONISTIC FUZZY SETS I Crme Bărbăcor, Cott Brâcş Uverty, Tg. J btrct: The rooed method ebe deco mker to chooe the mot mortt of the crter mkg deco g degree of memberh d o-memberh of the crter to the fzzy cocet mortce.. The rooed method e degree of tfbty d otfbty of ech tertve the trth-memberh fcto d o-trth memberh fcto. The robem tmtey e ovg er rogrmmg mode. The rooed method dffer from revo roche for mtcrter fzzy deco-mkg ot oy de to the fct tht the rooed method e ttotc fzzy et theory rther th fzzy et theory, bt o de to the degree of mortce of the crter re ot cott d the ccto mer. Key word: Ittotc Fzzy Set, Deco mkg, Memberh fcto, Degree of memberh, Degree of o-memberh, Degree of determcy.. INTRODUCTION Coderg the redctbe fctor deco-mkg, Zdeh [9] trodced the de of fzzy et whch h memberh fcto tht g to ech eemet of the vere of dcore, mber from the t terv [0,] to dcte the degree of beogge to the et der coderto. tov [] beqety rooed the cocet of ttotc fzzy et (IFS) by brgg o-memberh fcto together wth the memberh fcto of the fzzy et trodced erer by Zdeh [9]. mog the vro oto of hgher-order fzzy et, IFS rooed by tov[2] rovde febe frmework to eborte certty d vgee. Th de of IFS eem to be reorcef modeg my re fe tto ke egotto rocee, ychoogc vetgto, reog, medc dgo mog other.th er orgzed foow mog other Th er orgzed foow. I w frt reet the defto d roerte of ttotc fzzy et. I w cote to mtcrter deco-mkg method bed o ttotc fzzy et d the correodg er rogrmmg mode. Fy I w reet merc eme d hort coco. 2. INTUITIONISTIC FUZZY SETS The cocet of ttotc fzzy et w trodced by K.T. tov [] geerzto of the oto of fzzy et, X () ttotc fzzy et (IFS) X gve by,, X (2) 5

2 where μ : X [0,] ced degree of memberh d : X [0,] ced degree of o-memberh, wth the codto 0, X (3) We c degree of determcy of to, for ech X the mber [2]:, X (4) Gve two IFS d B over vere of dcore X, oe c defe the foowg reto: B ff X, B d B = B ff B d B we the foowg oerto []:,, X 2. MULTICRITERI FUZZY DECISION MKING BSED ON INTUITIONISTIC FUZZY SETS 2.. Preetto of the Probem I [3] we e the fzzy mtttrbte deco mkg roch to the roce of rodct eecto bed o qty of rodct, wch c eect the mot rorte oe wth the hghet degree of memberh beogg to the otve de oto. I thoght for mortt qty crter: rodcto cot (horty cot), tme, form, or. Etedg th hyothe, oe tht for yzed we hve et of m tertve, 2,..., m from whch yo mt eect oe, whch to ed the mber of crter to : C C,C.We we to evte 2 ech tertve C C. tfe the crter where B=, B, B X (5) B m, (6) B d B m, (7) B B=, B, B where X (8) B m, (9) B d B m, B (0) The degree of determcy re the, X. mber Ve the hgher hetto mrg of the deco mker of the me tht d Fgre. re the degree of memberh reectvey o-memberh of the tertve (Fgre ) whch tfy the crtero C C, reectvey, where 0 0 d, 0. I other word, the evto of the tertve wth reect to the crtero C ttotc fzzy et: C,, X () m, 6

3 tertve wth reect to the crtero C whoe tety gve by μ. Wth th de m to cree the qty of the emet roce. I [5], rooed to e octo the coed terv [, ] [, ]. Obvoy, 2 for d C C. I [5] the thor reeted () other form foow for the ke of erformg the deco-mker evto more drecty C,[, ] X. The thor trt from the mto tht there deco-mker who wt to chooe tertve whch tfe the crter C, C k,..., C or whch tfe the crter C. Bt rety mt chooe betwee two et of crter. So th er we me tht who wt to chooe tertve whch tfe the crter C,C k or whch tfe the crter C r,c t. Th deco-mker reqremet rereeted by the foowg ereo: (C ND C k ND... ND C )OR (C r ND C t ND... NDC ). (2) Therefore, the degree of evto the fct tht, whch the tertve tfe d doe ot tfy the deco-mker reqremet c be mered by the evto fcto F: C... C C... C F C C [, ], C C k... C, Cr C t... C [, ] [, ]... [, ] m,...,,m,..., m k r t k, k ]... [, ] r r t t, k, k r, t,...,,m r, t,...,, k r, t, k r, t [, ] [ [m m,...,,,...,, m m,...,,,..., ] [, ] m (ee retoh (5), (6), (7), (8), (9), (0)). [, ], where Let = [0,], [0,] 0,. The core of c be evted by the core fcto S how S( ), where S() [, ]. Net, [6] defe ccrcy fcto H to evte the degree of ccrcy of IFS where H() foow: H()= [0, ]. H(). Th ret the fct tht H hgher, the more the degree of ccrcy of the IFS. Ug the two fcto, [4] d the [5] coder fcto W, whch c mere the degree of tertve tfy the deco-mker reqremet. H E W E S E 2 3 W( E ) [, ]. The rger the ve of W(E( )), the more the tbty to whch the tertve tfe the decomker reqremet. 7

4 I [4] the thor reeted weghted techqe for hdg mtcrter fzzy deco-mkg robem, bt they med tht the degree of mortce of the crter etered by the deco-mker re cott, t hrd to do rety. I [5] the thor eted th hyothe d me tht ech crtero to hve dfferet degree of mortce, cty more commo rety. So, [5] the thor me tht d re the degree of memberh d omemberh of the crter C C to the fzzy cocet mortce, reectvey, where 0, 0 d 0. The thor coder ttotc dce = re ch tht the rger the hgher hetto mrg of deco-mker to the mortce of the crter C whoe tety gve by. Thee two ew qtte re ed to ccte the bgget weght (d the met oe) we c eect roce edg to f deco. Drg the roce the deco-mker c chge h evtg weght the foowg wy. It codered tht thee ve re cded the coed terv [, ] [, ]. Obvoy, 0 for ech crtero C C. I ddto, t med tht d order to fd otm weght tfyg d. metoed before, I w dd ddto to the [4] d [5], deco-mker who wt to chooe tertve whch tfe the crter C,C k or whch tfe the crter C r,c t. Th deco-mker reqremet c be rereeted by (2). The degree of mortce of the crter C,C k etered by the decomker re, k,...,, the degree of mortce of the crter C r,c t etered by the deco-mker re r, t,..., where,,, k k k, r r r t t t,,,..., d, k r t.... The weghtg fcto H d S wth degree of mortce, k,...,. Let [, ] [ k, k ] H [ ] T H * H * k..., * [, ] [ k, k ] S [ ] (3) W S * S * k..., * [ r, r ] [ t, t ] H [ ] (4) U H * H * r t..., * [ r, r ] [ t, t ] S [ ] (5) V S * S * r t..., * where [0,] W [,], [,] (6) U, T, [0,] V, m. The the degree of tbty tht the tertve tfe the deco-mker reqremet c be mered by the foowg fcto: 8

5 F T V m W, U (7) F, m. The rger the ve [,] of R( ), the more the tbty to whch the tertve tfe the deco-mker reqremet. I Eq. (7), we kow tht the ve of F( ) be o the ve of T, W,U, V. To obt weght, k,..., correodg crter C,C k d the weght r, t,..., correodg crter C r,c t [5] rooe to determe mmm fcto F g er rogrmmg. If t f to troe tht reqremet er rogrmmg robem the the oto w be obted g Sme method. The otm weght ve c be comted v er rogrmmg robem : The obectve fcto: 3 3 m F * * k 3 t 3t... * * t r 3 r 3 * r... * retrcto robem: (8)... (9) k... (20) r t o-egtvty codto:,,,, k k k (2) r r r, t t t,, (22) I d, th er rogrmmg robem oved g Sme method. 3. NUMERICL EXMPLE Net we w reet decomkg robem, where the chrctertc of the tertve re rereeted by ttotc fzzy et. Coder the robem of eecto of cr = {, 2, 3, 4, 5 }. The crter : C={c (rce), c 2 (comto), c 3 (tty) } re tke to coderto the eecto robem. Ug tttc method, the degree of memberh d the degree of omemberh for the tertve tfe the crtero c C c be obted, reectvey. Nmey,, 35 c 0.7,0.20.8,0.0.5,0.30.4, ,0.3 c2 0.5,0.30.9,0.0.7,0. 0.5, ,0.5 c3 0.8,0.20.6,0.30.7, , ,0. [, ] , , , , ,0.7 c 0.5, , , , ,0.5 c2 0.8, , , , ,0.9 c I mr wy, the degree ρ of memberh d the degree τ of omemberh for the three crter c C to the fzzy cocet mortce c be obted, reectvey, where =, 2, 3. Nmey,, , , ,0.55 Therefore, crter weght e the coed terv foow, [, ] , , ,0.45 9

6 ccordg to Eq. (8)-(22), the er rogrmmg c be obted:.975*.9* * 3 m F Ug Sme method to ove the bove er rogrmmg, t otm oto c be obted ω = 0.25, ω 2 = 0.3, ω 3 = The by yg Eq. (3), we c get 3 F5 0.6 * *0.4+ * * Therefore, we c ee tht the tertve 2 the bet choce. d the otm rkg order of the tertve gve by From the roce of ccto, we c ee tht the method reet th er eer th tht [4]. 4. CONCLUSION 3 F 0.7 * I th er t h bee yzed tht my deco mkg roch where the 3 3 *0.5+ * * chrctertc of the tertve re rereeted by ttotc fzzy et. The dfferece from other thor rem tht et of crter tht mt be ffed ech 3 tertve of the form (2). F2 0.8 * REFERENCE 3 3 *0.9+ * * [] tov, K.T., Ittotc fzzy et, Fzzy Set Syt., vo. 20, , gt 986. [Oe]. vbe: 3 htt://d.do.org/0.06/s065- F3 0.5 * (86) [2] tov, K.T., Ittotc fzzy et: 3 3 *0.7+ * * t, reet d ftre, EUSFLT Cof., M. Wgekecht d R. Hme, Ed. Uverty of ed Scece t Ztt/G ortz, Germy, 2003, [3] Bărbăcor, I. C., Fzzy Mt- F4 0.4 * ttrbte Deco Mkg gortm for 3 3 rodcto Comer Good, of the *0.5+ * * Cott Brâcş Uverty of Târg J- Egeerg Sere, o. 3, 203, , ISSN [4] Che S.M., T J.M., Hdg mtcrter fzzy deco-mkg robem bed o vge et theory, Fzzy Set Sytem 67(994) [8] Tog, H. Zhg, S., Fzzy Mt- ttrbte Deco Mkg gortm for Web Servce Seecto Bed o QoS, Proceedg of the 2006 IEEE -Pcfc 20

7 [5] L, L., Y, X.H., X Z.Q., Mtcrter fzzy deco-mkg method bed o ttotc fzzy et, Jor of Comter d Sytem Scece 73 (2007) [6] Che, S., Theory of fzzy eecto mttge d mtobectve deco mkg ytem, The Jor of Fzzy mthemtc, 994, 2(). [7] Che, S. J., Fzzy Mt-ttrbte Deco Mkg: method d cto, Ber: Srger-Verg, 992. Coferece o Servce Comtg, [9] L.. Zdeh, Fzzy et, Iform.d Cotro 8 (965)