Saw-Dmss Model For Intuitionistic Fuzzy Multi Attribute Decision Making Problems

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1 Itratoal Joural o Rct ad Iovato Trds Computg ad Commucato ISSN: Saw-Dmss Modl For Itutostc Fuzzy Mult ttrbut Dcso Mag Problms V. Thagarasu ssocat Profssor of Computr Scc Gob rts & Scc Collg Gobchttpalayam, Ida N. Thahara ssstat Profssor Computr Scc Thatha Has Rovr Collg Prambalur, Ida t.msu203@gmal.com bstract Ths wor troducs a w SW-DMSS (Smpl ddtv Wghtg-Dcso Mag Support Systm) tchqu for dcso-mars to choos th most dal altratv that has b provdd. Ths also dals wth th problm basd o SW algorthm whch s a multpl crtra dcso mag approach wth wght dtrmg mthods whch gvs th wghts to dcators whch s partally or compltly uow or ot prstd by th dcso mars. Th SW algorthm dals wth th coflcts btw dcators basd o crta way to sort th schm ad choos th bst schm. umrcal xampl s proposd to llustrat th ffctvss of ths algorthm. Howvr, comparso of two wght dtrmg mthods basd o Gaussa dstrbuto ad Lgustc quatfr gudd aggrgato s prformd to ma th rsult of valuatos mor obctv ad accurat. Kywords-SW; DMSS; Dcso Mag; Mult Crtra Dcso Mag. ***** I. INTRODUCTION Dcso-mag support systms (DMSS) ar computr basd formato systms dsgd to support som or all phass of th dcso-mag procss. Dcso-mag support systms utlz cratv, bhavoral, ad aalytc foudatos that draw o varous dscpls. DMSS voluto has prstd uqu challgs ad opportuts for formato systm profssoals. Ths foudatos gv rs to varous archtcturs that dlvr support to dvdual ad group DMSS usrs. Oc cratd, DMSS must b valuatd ad maagd. Ecoomc-thory-basd mthodologs, quattatv ad qualtatv procss ad outcom masurs, ad th dashboard approach hav b usd to masur DMSS ffctvss. Ths wor dals wth th DMSS problms basd o SW algorthm (Smpl ddtv Wghtg) whch s a multpl crtra dcso mag approach wth tutostc fuzzy sts. Th SW algorthm dals wth th coflcts btw dcators basd o crta way to sort th schm ad choos th bst schm. Som valus of th mult attrbut dcso modls ar oft subctv. Th wghts of th crtra ad th scorg valus of th altratvs agast th subctv (udgmtal) crtra cota always som ucrtats. It s thrfor a mportat qusto how th fal rag or th rag valus of th altratvs s sstv to th chags of som put paramtrs of th dcso modl I multpl attrbut dcso mag (MDM) problm, a dcso mar (DM) has to choos th bst altratv that satsfs th valuato crtra amog a st of caddat solutos. It s grally hard to fd a altratv that mts all th crtra smultaously, so a bttr soluto s prfrrd. Th SW mthod was dvlopd for multcrtra optmzato of complx systms. Ths mthod focuss o rag ad slctg from a st of altratvs th prsc of coflctg crtra. Mult-crtra optmzato s th procss of dtrmg th bst fasbl soluto accordg to th stablshd crtra (rprstg dffrt ffcts). Practcal problms ar oft charactrzd by svral ocommsurabl ad coflctg crtra ad thr may b o soluto satsfyg all crtra smultaously. Thus, th soluto s a st of o-fror solutos, or a comproms soluto accordg to th dcso mar s prfrcs. Th comproms soluto was stablshd by Zly, (982) for a problm wth coflctg crtra ad t ca hlp th dcso mars to rach a fal soluto. I classcal MDM mthods, th ratgs ad th wghts of th crtra ar ow prcsly, whras th ral world, a mprcs ad ucrta vromt, t s a uralstc assumpto that th owldg ad rprstato of a dcso mar or xprt ar so prcs. For xampl, huma udgmt cludg prfrcs s oft vagu ad dcso mar (DM) caot stmat hs prfrc wth xact umrcal valus. I ths stuatos, dtrmg th xact valu of th attrbuts s dffcult or mpossbl. So, to dscrb ad trat mprcs ad ucrta lmts prst a dcso problm, fuzzy approachs ad lgustc trms ar frqutly usd. I th wors of lgustc trms dcso mag, lgustc trms ar assumd to b wth ow by fuzzy lgustc mmbrshp fucto. Howvr, ralty to a dcso mar t s ot always asy to spcfy th mmbrshp fucto a xact vromt. t last som of th cass, th us of trval umbrs may srv th purpos bttr. trval umbr ca b thought as a xtso of th cocpt of a ral umbr, howvr, dcso problms ts us s ot much attdd as t mrts (Hwag & Yoo, 98). Thagarasu & Thahara, (205) ad Thagarasu & Rgara, (205) hav cotrbutd to th fld of DMSS usg SW ad VIKOR mthods. Ch, (202) prstd a comparatv modl basd IJRITCC May 207, 364

2 Itratoal Joural o Rct ad Iovato Trds Computg ad Commucato ISSN: o SW ad TOPSIS. Zavadsas t al., (2007) prstd a sstvty aalyss for SW mthod. Vaguss ad ucrtaty ar th two mportat aspcts of mprcso. IFS s a tutvly straght forward xtso of Zadh s, (965) fuzzy sts. IFS thory bascally dfs th clam that from th fact that a lmt x blogs to a gv dgr (say μ) to a fuzzy st, t aturally follows that x should ot blog to to th xtt, a assrto mplct th cocpt of a fuzzy st. O th cotrary, IFSs assgs to ach lmt of th uvrs both a dgr of mmbrshp ad o of o-mmbrshp such that, thrby rlaxg forcd dualty from fuzzy st thory. Obvously, wh for all lmts of th uvrs, th tradtoal fuzzy st cocpt s rcovrd. I IFS ths dtty s wad to a qualty, or othr words: a dal of th law of th xcludd mddl occurs, o of th ma das of tutosm. Lt X b th X x, x,..., x. Th uvrs of dscours dfd by 2 grad of mmbrshp of a lmt x X a fuzzy st s rprstd by ral valus btw 0 ad. It dcats th vdc for x X, but dos ot dcat th vdc agast x X. taassov, (986; 989 ) potd out that ths sgl valu combs th vdc for x X ad th vdc agast x X. IFS X s charactrsd by a mmbrshp fucto ( x ) ad a o-mmbrshp fucto ( x ). Hr, ( x ) ad ( x ) ar assocatd wth ach pot X, a ral umbr [0,] wth th valus of ( x ) ad ( x) at X rprstg th grad of mmbrshp ad o-mmbrshp of x. Thus closss of th valu of ( x) to uty ad th valu of ( x) to zro, ras hgh th grad of mmbrshp ad lowr th grad of o-mmbrshp of x. IFS bcoms a fuzzy st wh ( x) 0. Cossus procsss mply that xprts achv a agrmt about a problm bfor tag a dcso, thus yldg a soluto accptd by th orgazato, socty or thmslvs. I ths approachs, t s crucal to stablsh a cossus masur to calculat th lvl of agrmt. Cossus masurs ar dcators to valuat how far a group of xprts opos s from uamty. Mohaty & Bhasr, (2005), Mohaty & Zahr, (2007) ad Mohaty, (2008) hav appld th cocpts of Lgustc Quatfrs th product classfcatos basd o customr prfrc Itrt- Busss. I ths papr, Th RIM-Lgustc Quatfrs (Yagr, 988) basd o th Ordrd Wghtd vragg (OW) oprators ar usd to drv th wghts of th xprts ad Gaussa Dstrbuto basd mthod proposd by Xu, (2005) s also usd for th sam purpos. algorthm for th proposd modl of MDM for tutostc fuzzy sts s prstd ths chaptr. umrcal llustrato s prstd wh th wghts ar uow. comparso of th proposd mthods s also prstd. II. PPLICTION OF SW S DECISION SUPPORTSYSTEM TECHNIQUE DMSS s tdd to support, rathr tha rplac, dcso mar s rol solvg problms. Dcso mars capablts ar xtdd through usg DSS, partcularly ll-structurd dcso stuatos. I ths cas, a satsfd soluto, stad of th optmal o, may b th goal of dcso mag. Solvg ll-structurd problms oft rls o rpatd tractos btw th dcso mar ad th DSS. Dcso support systms ar bult upo varous dcso support tchqus, cludg modls, mthods, algorthms ad tools. cogto-basd taxoomy for dcso support tchqus, cludg sx basc classs as follows: Procss modls, Choc modls, Iformato cotrol tchqus, alyss ad rasog tchqus, Rprstato ads ad Huma udgmt amplfyg/rfg tchqus. Th Multcrtra dcso mag ad Mult-attrbut dcso mag coms udr th catgory of Choc modls. Multpl ttrbut dcso support systms ar provdd to assst dcso mars wth a xplct ad comprhsv tool ad tchqus ordr to valuat altratvs trms of dffrt factors ad mportac of thr wghts. Som of th commo Mult-ttrbut Dcso-Mag (MDM) tchqus ar: Smpl ddtv Wghtd (SW) Wghtd Product Mthod (WPM) Coopratv Gam Thory (CGT) IJRITCC May 207, Tchqu for Ordr Prfrc by Smlarty to Idal Soluto (TOPSIS) Elmato t Choc Traslatg Ralty wth complmtary aalyss(electre) Prfrc Rag Orgazato Mthod for Erchmt Evaluato (PROMETHEE) alytcal Hrarchy Procss (HP) Th mrt of th SW mthod s that t ca dal wth both quattatv ad qualtatv assssmt th procss valuato wth lttl computato load. It bass upo th cocpt that th chos altratv s drvd from th wghtd dcso matrx. I th procss of SW, th prformac ratgs ad th wghts of th crtra ar gv as crsp valus. I fuzzy SW, attrbut valus ar rprstd by fuzzy umbrs.. SW mthod Dcso-mag problm s th procss of fdg th bst opto from all of th fasbl altratvs. I almost all such problms, th multplcty of crtra for udgg th altratvs s prvasv. For may such problms, th DM wats to solv a multpl attrbut dcso mag (MDM) problm (Hwag & Yoo, 98). MDM problm ca b cocsly xprssd matrx format as: C x x... x 2 x x... x m x x... x m m2 m whr, 2,..., m ar possbl altratvs amog whch dcso mars hav to choos, C,C 2,...,C ar crtra wth whch altratv prformac ar masurd, x s th ratg of altratv wth rspct to crtro C. 365

3 Itratoal Joural o Rct ad Iovato Trds Computg ad Commucato ISSN: SW Tchqu s o of th most usd MDM B ( x) B( x) ad ( x) B( x), x X. tchqu. It s smpl ad s th bass of most MDM tchqus such as HP ad PROMETHEE that bfts from addtvs. I SW tchqu, fal scor of ach altratvs s C. OPERTIONS DEFINED OVER INTUITIONISTIC calculatd as follows ad thy ar rad. FUZZY SETS: P w. r ;,2,..., m. Whr r ar ormalzd valus of dcso matrx lmtsad calculatd as follow: For proft, attrbuts, w hav, r max d ; d Max d ;,2,..., d for cost attrbuts, r m d ; d M d ;,2,..., If th thr s ay qualtatv attrbutv, th w ca us som mthods for trasformg qualtatv o s. B. INTUITIONISTIC FUZZY SETS Lt Xb th uvrs of dscours. tutostc fuzzy st Xs a obct havg th form x, ( x), ( x) x X whr ( x), ( x): x [0,] dot mmbrshp fucto ad ommbrshp fucto, rspctvly, of ad satsfy 0 ( x) ( x) for vry x X. ( x ) s th lowst boud of mmbrshp dgr drvd from proofs of supportg x; ( x ) s th lowst boud of o-mmbrshp dgr drvd from proofs of rctg x. It s clar that th mmbrshp dgr of Itutostc Fuzzy st has b rstrctd [ ( x), ( x)] whch s a subtrval of [0,]. For ach IFS X w call ( x) ( x) ( x) as th tutostc dx of x. It s hstato dgr (or dgr of dtrmacy) of x to. It s obvous that 0 ( x ) for ach x X. For xampl, lt b a IFS wth mmbrshp fucto ( x ) ad o-mmbrshp fucto ( x ), rspctvly. If ( x ) 0.5 ad ( x ) 0.3, th w hav ( x ) It could b trprtd as th dgr that th obct x blogs to th IFS s 0.5, th dgr that th obct x dos ot blog to th IFS s 0.3 ad th dgr of hstato s 0.2. Thus, th IFS X ca b xprssd as x, ( x), ( x), ( x) : x X If s a ordary fuzzy st, th ( x) ( x) ( ( x)) 0 for ach x X. It mas that th thrd paramtr ( x ) caot b casually omttd f s a gral IFS, ot a ordary fuzzy st. Thrfor, th rprstato of IFS should cosdr all thr paramtrs calculatg th dgr of smlarty btw IFSs. For, B IFS( X ), th st of all IFSs, th oto of cotamt s dfd as follows: Frst, Som opratos,,, ar dfd ovr tutostc fuzzy sts (IFSs). Hr w shall dscuss som of thr basc proprts. For vry two IFSs ad B th followg ar vald (lt, [0,]) : IJRITCC May 207, B ff ( x E) ( x) ( x) & ( x) ( x), B ff B, B B B ff ( x E) ( x) ( x) & ( x) ( x), x, ( x), ( x) x E, B B B x,m( ( x), ( x)), m( ( x), ( x)) x E, B B B x,m( ( x), ( x)), m( ( x), ( x)) x E, B B B x, ( x) ( x) ( x) ( x), ( x) ( x)) x E, B B B B x, ( x) ( x), ( x) ( x) ( x) ( x)) x E, B B B D. GUSSIN METHOD OF DETERMINING UNKNOWN WEIGHTS Lt us cosdr a stuato whr thr s a ufar argumt amog th xprts fxg th wghts a dcso mag problm. I that cas w d to rlv th fluc of ufar argumts o th dcso varabls. Xu, (2005) troducd a procdur for gratg th wghts basd o th us of th Gaussa dstrbuto. Thy ar rfrrd as Gaussa wghts whch ar gv as follows: Cosdr a Gaussa dstrbuto G(, ), whr s th ma of th collcto ad s th dvato of th collcto, ad gv by: ad 2 Lt G( ) 2 ( ). 2 ( ) /2. Th th assocatd wghts ar dfd as: ( ) /2 G w = ( ) /2 G( ) whr w [0,] ad w. It ca b otd that th closr s to, th largr 2. Furthrmor, f s odd, th maxmal valu of w occurs w 366

4 Itratoal Joural o Rct ad Iovato Trds Computg ad Commucato ISSN: xampl of RIM quatfr Q = most, wth a = 0.5 ad b for. If s v, th maxmal valu of w occurs 2 = 0.7 s gv as: for ad. It ca also b show that th 0 f r 0.5 wghtg vctor gratd usg ths approach s symmtrc, Qr 5r 2.5 f 0.5 r 0.7.., w. f r 0.7 w E. LINGUISTIC (RIM) QUNTIFIERS FOR DETERMINING UNKNOWN WEIGHTS Th problm of dtrmg wghts for a OW oprator ca b addrssd dffrt ways, for xampl wth th us of th so-calld Lgustc Quatfrs. rlatv lgustc quatfr Q, such as most, fw, may, ad all, ca b rprstd as a fuzzy subst of th ut trval, whr for a gv proporto r 0, of th total of th Sc th us of OW wth RIM quatfrs capturs th oto of th soft cossus corrctly, thy ca b adoptd for th purpos of studyg th ffct of dffrt aggrgato oprators o th rsoluto of a cossus problm wth may xprts, ad xprssg a dsrd group s atttud. Fgur Rprstato of th Lgustc Quatfr Q(r) valus to aggrgat, Q(r) dcats th xtt to whch ths proporto satsfs th smatcs dfd Q. For xampl, gv Q = most, f Q(0.7) = th t would ma that a proporto of 70% totally satsfs th da covyd by th quatfr most, whras Q(0.55) = 0.25 dcats that th proporto 55% s barly compatbl wth ths cocpt (.., oly 25%). Rgular Icrasg Mooto (RIM) quatfrs ar spcally trstg for thr us OW oprators. Ths quatfrs prst th followg proprts:. Q(0) = 0. Q() =. If r r th Qr Qr. 2 2 Yagr, (988) suggstd th followg mthod to comput wghts w, wth th us of a RIM quatfr Q: w Q Q,,2,...,. Whr th mmbrshp fucto of a lar RIM quatfr Q(r) s dfd by two paramtrs ab, 0, as: 0 f r a r a Qr f a r b b a f r b F. LGORITHM FOR INTUITIONISTIC FUZZY SW- DMSS Th followg stps ar followd for th tutostc fuzzy SW-DMSS proposd ths papr: G. Stp-: For th dcso matrx R fd th dfuzzyfd matrx ad th ormalzd matrx. Stp-2: Calculat th wghts of th attrbuts by Gaussa mthod ad lgustc quatfr mthod. Stp-3: For th xpctd attrbut valu matrx R, calculat th wghtd ormalzd matrx. Stp-4: Calculat th xpctd valu from th wghtd ormalzd matrx. Stp-5: Ra th altratvs ad choos th bst o accordg th rag ordr. III. NUMERICL ILLUSTRTION W assum a MDM problm that has thr altratvs ad four attrbuts whr attrbuts C, C 4 ar cost typ ad attrbuts C 2,C 3 ar of proft typ. Th tutostc fuzzy dcso matrx s gv as follows: (0.29,0.9) (0.337,0.568) (0.3809,0.904) (0.465,0.2307) D (0.45,0.35) (0.5490,0.2745) (0.5079,0.2539) (0.3590,0.795) (0.48,0.24) (0.4706,0.2353) (0.4444,0.2222) (0.528,0.2564) Th dfuzzyfd matrx from th abov tutostc fuzzy matrx s gv as follows: IJRITCC May 207, 367

5 Itratoal Joural o Rct ad Iovato Trds Computg ad Commucato ISSN: D Th ormalzd matrx s gvs as: P Wght Dtrmato by Gaussa Dstrbuto Mthod: Th thr possbl altratvs of th abov dcso mag problm ar to b valuatd by thr dcso mars whos wghtg vctor s compltly uow. Th ma ad th dvato of th collcto, 2,, ar gv by th followg quatos as follows: Whr 2 ad 2,.5, , 0.5, Calculatg th xpctd valu from th wghtd ormalzd matrx, w gt: P r w,,2,..., m P r w 0.763, P2 r w.2282, P3 r w Th rag of th abov thr altratvs wll gv th bst altratv: P2 PP3. Hc th bst altratv s P 2 (Scod altratv). Th th wghts ar calculatd usg th followg quato as follows: w G = G whr w [0,] ad w. 2 2 w = , w 2 = 0.542, w 3 = Hc th wghts of th xprts ar ta as V = (0.2429, 0.542, ) T. Usg th wght vctor ad procdg wth th abov wghtd ormalzd matrx, w gt: Th wghtd ormalzd matrx s gv as follows: P w B. Wght Dtrmato by Lgustc Quatfr Mthod: Th thr possbl altratvs of th abov dcso mag problm ar to b valuatd by thr dcso mars whos wghtg vctor s compltly uow. Th mmbrshp fucto for th lgustc quatfr Q = most s gv as follows: most most 0 f x 0.5 x 0.5 f 0.5 x f x f x 0.5 2x 0.4 f 0.5 x 0.9 f x 0.9 Th uow wghts ar computd by th RIM quatfr Q as follows: w Q Q,, 2,...,. Whch gvs th wghts as w = (0.26, 0.68, 0.06) T. Usg ths wght vctor drvd from th RIM lgustc quatfr, ad IJRITCC May 207, 368

6 Itratoal Joural o Rct ad Iovato Trds Computg ad Commucato ISSN: procdg wth th abov wghtd ormalzd matrx, w Mthod-2: Uow gt: Exprt Wghts P2 P P3 (RIM Lgustc. Quatfr) Th wghtd ormalzd matrx s gv as follows: IV. CONCLUSION: P w Calculatg th xpctd valu from th wghtd ormalzd matrx, w gt: P r w,,2,..., m P r w 0.849, P2 r w.6243, P3 r w Th rag of th abov thr altratvs wll gv th bst altratv: P2 PP3. Hc th bst altratv s P 2 (Scod altratv). Tabl- Proposd SW-DMSS modls wth uow wghts SW-DMSS Modls Mthod-: Uow Exprt Wghts (Gaussa Dstrbuto) Rag of ltratvs P2 P P3. FINDINGS ND SUGGESTIONS Th proposd rsarch wor has coctratd o applyg SW-DMSS mthod to ral world dcso mag problms. Th gral SW-DMSS mthod was proposd ad w algorthm was proposd for Multpl ttrbut Dcso Mag ffctly. umrcal llustrato wth th thory of slctg th bst altratv s aalyzd wth th hlp of th proposd algorthm of SW-DMSS mthod xtdd wth applyg th chags tag plac dtrmg uow xprt wghtg vctor s prstd. Th umrcal llustrato prstd utlzg th SW-DMSS mthod dsplays th ffctvss of th proposd algorthm. REFERENCES [] taassov, K., (986). Itutostc fuzzy sts. Fuzzy Sts ad Systms, 20, [2] taassov, K., (989). Mor o tutostc fuzzy sts. Fuzzy Sts ad Systms, 33, [3] Ch, T-Y. (202). Compartv aalyss of SW ad TOPSIS basd o trval-valud fuzzy sts: Dscussos o scor fuctos ad wght costrats. Exprt Systms wth pplcatos, 39 (2), [4] Chg, S.K., (2000). Dvlopmt of a Fuzzy Mult-Crtra Dcso Support Systm for Mucpal Sold Wast Maagmt. mastr thss of appld scc dvacd Maufacturg ad Producto Systms, Uvrsty of Rga, Sasatchwa. [5] Hwag, C. L., & Yoo, K. (98). Multpl attrbut dcso mag mthods ad applcatos: stat of th art survy. Nw Yor: Sprgr- Vrlag. [6] Mohaty, B.K., & Bhasr, B., (2005). Product Classfcato th Itrt Busss- Fuzzy pproach. Dcso Support Systms, 38, [7] Mohaty, B.K., & Zahr, S., (2007). Maagg E- Busss Customr Focussd- Fuzzy pproach. Maagg Worldwd Opratos & Commucatos wth Iformato Tchology, 2007 IRM Itratoal Cofrc, [8] Mohaty, B.K., (2008). Traqulty ad xty E-Busss- Fuzzy pproach Itratoal Cofrc o dvacd Computr Thory ad Egrg, IEEE Computr Socty, 8-2. [9] Thagarasu, V., & Thahara, N. (205). MDM modl for dcso mag support systm usg SW mthod, Itratoal Joural of Iformato Scc ad Itllgt Systm, 4(), [0] Thagarasu, V., & Rgara, V. (205). MDM modl wth VIKOR mthod for dcso mag support systms, Itratoal Joural of Novl Rsarch Computr Scc ad Softwar Egrg, 2(), [] Xu, Z.S., (2005). Ovrvw of Mthods for Dtrmg OW Wghts. Itratoal Joural of Itllgt Systms, 20, [2] Yagr, R.R., (988). O ordrd wghtd avragg aggrgato oprators multcrtra dcso mag. IEEE Trasactos o Systms, Ma, ad Cybrtcs, 8, [3] Zadh, L.., (965). Fuzzy Sts. Iformato ad Cotrol, 8, [4] Zavadsas, E. K., Turss, Z., Dus, T., &Vt, M. (2007). Sstvty aalyss of a smpl addtv wght mthod. Itratoal Joural of Maagmt ad Dcso Mag, 8(5), [5] Zly, M., (982). Multpl crtra dcso mag. Nw Yor: McGraw-Hll. IJRITCC May 207, 369