A Solution for multi-evaluator AHP

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Peter Newman
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1 ISAHP Honoll Hw Jly 8- A Solton for lt-vltor AHP Ms Shnohr Kch Osw Yo Hd Nhon Unvrsty Nhon Unvrsty Nhon Unvrsty Iz-cho Nrshno Iz-cho Nrshno Iz-cho Nrshno hb 7-87 Jpn hb 7-87 Jpn M7snoh@ct.nhon-.c.p 7oosw@ct.nhon-.c.p hb 7-87 Jpn 7snoh@ct.nhon-.c.p Kywords: lt-vltor vltor s wght logrthc lst sqrs stton Sry: W oftn ncontr th lrg scl AHP whr thr r ny nds of ltrntvs or obcts to b vltd nd on vltor cnnot covr whol obcts so svrl sprt vltors r ndd frthr ch vltor hs th spcfc blty to vlt spcfc grop of obcts. Lt sch typ of AHP b clld lt-vltor AHP. o solv th lt-vltor AHP w propos logrthc lnr prws coprson rror odl tng vltor's spcfc chrctrstc n consdrton nd thn th lst sqr prncpl s ppld to obtn stts of obct wght nd vltor wght. h physcl nng of wghts of vltors s clrfd by th spcfc forl obtnd n or nlyss. Frthr th so clld grop dcson ng n AHP s spcl cs of lt-vltor AHP by whch w cn vlt th rlblts of vltors.. Introdcton W oftn ncontr th lrg scl AHP[] whr thr r ny nds of ltrntvs or obcts to b vltd nd on vltor cnnot covr whol obcts so svrl sprt vltors r ndd frthr ch vltor hs th spcfc blty to vlt spcfc grop of obcts. Lt sch typ of AHP b clld lt-vltor AHP. o solv th lt-vltor AHP w propos logrthc lnr prws coprson rror odl tng vltor's spcfc chrctrstc n consdrton nd thn th lst sqr prncpl s ppld to obtn stts of obct wght nd vltor wght. Alrdy K. t l[] proposd solvng thod for ths typ of probl by ANP[] whch gvs wll rsonbl lgorth. Or pproch s dffrnt fro thrs nd gvs th followng spcfc profts whch r not sn n [] ; h physcl nng of wghts of vltors s clrfd by th spcfc forl obtnd n or nlyss. Frthr th so clld grop dcson ng [][] n AHP s spcl cs of lt-vltor AHP by whch w cn vlt th rlblts of vltors.. h rror odl of lt-vltor AHP Fgr shows spl odl of lt-vltor AHP whr thr r vltors

2 AB nd obcts. A tchs obcts "" B tchs obcts "" nd tchs obcts "". Ech vltor gvs prd coprson vls for th obcts of hs grop l th ordnry AHP. Hr lt b c b th prd coprson vl for obcts nd gvn by A B nd b th tr vl of obct. Ech vltor s consdrd to hv dffrnt vltng crtron nd lt αβγ b th wght of AB for chrctrzng th vltor's tttd thn w ss tht α β b γ c ns pproxton. hs stton s lso dscrbd by th grph shown n Fgr. Lt t b clld grph of lt vltor AHP. Any connctd grph wth wghtd drctd rcs cn b grph of lt vltor AHP s w s ltr xpls. Not tht rc orgn nd pont wth α wght α corrsponds to n th bsc forl ng logrth of w hv α b β c γ

3 whr x log x for x tc. W pply th prncpl of lst sqrs LS to to L -nor for th rror n nl n α β. α } { b β } { c S{ γ } h nzng α β γ s th stt of th vl of th wght of AB nd th nzng s th stt of th tr vl of obct. Hr w ss th followng constrnts ; α β γ h nbr of vltors. Gnrlly nd r fr fro constnt ltpl so w cn ss whos logrth s. W cn ccpt bcs th stndrd vls of vltor wghts r.. Solvng thod of lt-vltor AHP Alrdy w hv dscrbd th solvng prncpl n hptr. Hr w forlz th prncpl by th nw sybols to solv gnrl lt-vltor AHP nd frthr dscrb svrl proprts of th solton. Frstly w ntrodc svrl sybols ; υ :wght αυ of vltorυ υ υ αυ :logrth of th tr vl of obct :logrth of th prd coprson vl of th vltor corrspondng to rc n th grph of lt-vltor AHP n.h ordr of th nbrng of s th followng ; th st of rcs r dcoposd nto grops of vltor υ υ nd rc wthn grop υ s spcfd by υ so th ordr of s dtrnd lxcogrphclly by υ. For xpl th grph n Fgr s rwrttn by nw sybols n Fgr

4 Now th LS prncpl s to nz n S ' υ ndr th condtons 7 nd 8 whr ' υ s dtrnd by. nd 7 8 Expl W wrt down 7 8 for th lt vltor AHP shown n Fgr. n } } { S So th probl s n LS probl nzng wth constrnts 7 nd 8. hs cn b solvd by Lgrng thod. Lt th Lgrng ltplrs of 7 nd 8 b λ nd rspctvly nd thn

5 .. S L λ. hn th solton of or probl s obtnd by solvng th qtons L 9 υ L υ wth 7 nd 8. ng th root of ch tr n to b rror w hv υ n whch s rprsntd by th trx for whr n nd s th ncdnc trx of th grph of lt-vltor AHP nd υ s th vctor whos coponnts r prd coprson dt by vltor υ υ. bl : Dt bl Dt bl of Expl

6 Forl wth ottng rror tr s oftn rprsntd by tbl clld dt tbl shown n bl. Lt th coffcnt trx of Dt bl to b X X thn th norl qton of ths LS probl s gvn by whr nd r ll coln vctors of dnson nd rspctvly X X X M M λ And th soltons of

7 [ ] [ ] λ r or dsrd rslts. Expl onsdr lt-vltor AHP shown n Fgr. For ths xpl w hv X nd th norl qton of s shown n bl. bl : Norl qton of lt-vltor AHP λ For

8 th solton s gvn s blow.7..8 α α.9.9 λ.88.. Physcl nng of vltors wght Dcoposng nto -prt nd -prt by w hv Τ - λ nd û 7 8 Hr s -th dcoposd prt corrspondng to vltor of tht s. 9 By 7 nd 8 w hv th followng thors. hor h LS solton λ of Lgrng ltplr λ for 7 s lwys zro. Proof It s clr tht. bcs s th ncdnc trx of th grph of lt-vltor AHP. So w hv lso. Mltplyng 7 fro th lft by w hv whch shows λ. λ hor h LS solton of Lgrng ltplr for 8 s Proof Dvdng 8 by û. nd sng p on w hv by 8

9 û whch lds s to. rnsltng α to α w hv by 8 û - α. Forl hs vry portnt physcl nng of vltor whch w xpln throgh th followng xpl. Expl onsdr th lt-vltor AHP n Fgr. h dcoposton of s gvn s blow If th coprson dt of vltors r ll xct tht s 9 thn gnrlly û hn by hor w hv. W cn sy tht s nd of gnrl crtron to sr th ccrcy of vltors. { } { } { } α α α

10 Whn α û s nd of corrlton coffcnt of coprson dt nd corrspondng n vltor. So th vltor hvng α nr to s rlbl vltor. If vltor hs tndncy of ndrstts for prd coprsons δ û - bcos postv so α û δ bcos lrgr thn. h contrry cs s lso vld nd s rslts w cn sy ; vltor hvng α s rlbl on nd hvng α > < hs tndncy of ndrovrstts for prd coprsons. And frthr by hor w cn sy; If t shows tht s whol vltors r rlbl nd >< shows tht vltors s whol hv tndncy of ndrovrsttng. h so clld "grop dcson probl"gdp n AHP s spcl cs of lt-vltor AHP. GDP trts th probl whr vry vltor covrs ll obcts. A spl cs of GDP s shown n Fg.. For th nlyss of GDP w hv only to t ql to ncdnc trx of th grph of lt-vltor AHP n 8 nd 9. For xpl th dt tbl for Fg. s shown n bl. W cn s th dplcts of n th coln of -prt. bl : Dt bl of GDP Expl

11 - - onvntonl nlyss of GDP s to t th gotrc n of th rslts of vltors bt by ths convntonl nlyss w cnnot s th vltors blts. Howvr w cn clrfy th vltors wght α by th bov sttnts. For nd..87 th soltons s gvn s blow α.7. α λ -.8. onclson nd frthr rsrch W proposd solvng thod of lt-vltor AHP bsd on th constrnd lst sqrs LS thod whch gvs not only th stts of wghts of ltrntvs bt lso thos of vltors hptr. Frthr by or thod th physcl nngs of vltor s wght s clrfd throgh Lgrng ltplr for th constrnt. W cn solv th grop dcson probls n AHP by or thod s spcl cs of lt-vltor AHP hptr. W wold l to xtnd lt-vltor AHP to dsgn probls whch nfor ch vltor to slct wht sbst of whol obcts to vlt. Rfrncs [].L.Sty :h Anlytc Hrrchy ProcssMc-Grw Hll98 [].L.Sty :h Anlytc Ntwor ProcssRWS-Pblcton99 []K. Y.Sgy nd H.r :A Grop Anlytc Ntwor ProcssANP for Incoplt nfortc Syst Sgyo Joho G-n Jpns []M.Nnsh nd E.Knosht :An Applcton of th Grop Dcson Mng Strss Mthod to th Grop Anlytc Hrrchy Procss Jornl of h Oprtons Rsrch Socty of Jpn998-7n Jpns []Yoshys Yd. Mnb Sgy nd Noz Y Grop Anlytc Hrrchy Procss Bsd on onsnss Mng Modl Jornl of h Oprtons Rsrch Socty of Jpn 997 ~ n Jpns

Convergence Theorems for Two Iterative Methods. A stationary iterative method for solving the linear system: (1.1)

Convergence Theorems for Two Iterative Methods. A stationary iterative method for solving the linear system: (1.1) Conrgnc Thors for Two Itrt Mthods A sttonry trt thod for solng th lnr syst: Ax = b (.) ploys n trton trx B nd constnt ctor c so tht for gn strtng stt x of x for = 2... x Bx c + = +. (.2) For such n trton