펦폃쫽톧MATHEMATICAAPPLICATA2001,14(2):61ꆫ63Ξ폃AHP랽램럖컶슢뛏뺺헹쫐뎡췴볒뿛,쇖ꆡ뒨(훐맺뿆톧벼쫵듳톧쫽톧쾵,낲믕뫏럊230026)햪튪:놾컄폃AHP랽램퇐뺿쇋슢뛏뺺헹쫐뎡뗄탔훊,횸돶슢뛏뺺헹쫐뎡뗄뻹뫢볛룱뻍쫇훊솿뇈뷏뻘헳뗄훷쳘헷쿲솿,늢퓚듋믹뒡짏,럖컶쇋짺닺헟쪵쿖샻죳벫듳뮯뗄쳵볾.맘볼듊:뇈뷏뻘A;ퟔ컒움뚨볛룱;뿍맛움뚨볛룱;뻠샫몯쫽;뻹뫢볛룱훐춼럖샠뫅:ꆡꆡꆡAMS(2000)훷쳢럖샠:90A12;90A60컄쿗뇪쪶싫:Aꆡꆡꆡ컄헂뇠뫅:100129847(2001)0220061203쫐뎡뺭볃훐듦퓚듳솿슢뛏뺺헹탐튵,쯼훷튪폐틔쿂벸룶쳘뗣:(1)쒳탐튵듦퓚듳솿뗄짺닺헟뫍쿻럑헟,짺닺헟횮볤듦퓚벤쇒뺺헹;(2)늻춬짺닺헟뗄닺욷듦퓚훊솿뫍볛룱짏뗄닮뇰;(3)짺닺헟뿘훆닺욷뗄볛룱뫍닺솿.짨짺닺헟벯뫏I={1,2,ꆭn}.뛔헢n룶짺닺헟뗄닺욷훊솿뷸탐쫐뎡뗷닩,죃맣듳쿻럑헟샻폃돉뛔뇈뷏램뛔헢킩닺욷뷸탐훊솿뇈뷏,뿉틔뗃떽튻훊솿뇈뷏뻘헳A=(aij)nꇁn,웤훐aij>0,aii=1,i,j=1,2,ꆭ,n,쇮Aδ=(A-I)ö(n-1),Aδ=(aδij)nꇁn.쿔좻폐aδii=0,i=1,ꆭ,n,볇A뗄뗚i탐쿲솿캪ai,Aδ뗄뗚i탐쿲솿캪aδ.i틗횪A,Aδ뻹쫇럇뢺늻뿉풼랽헳.짺닺헟i뛔ퟔ벺닺욷뗄뚨볛돆캪ퟔ컒움뚨볛룱,볇캪Pi,튲벴쫇쫐뎡짏룃닺욷뗄쿺쫛볛룱.쇮δpi=aδip,돆δpi캪뿍맛움뚨볛룱,쯼뇭쪾쿻럑헟퓚늻횪뗀닺욷i뗄볛룱뗄쟩뿶쿂,춨맽폫웤쯼닺욷훊솿뫍볛룱짏뗄뇈뷏,뛔룃닺욷뗄볛룱맀볆,쯼뿍맛뗘랴펳쇋룃닺욷뗄볛룱쮮욽.쇮ri(p)=aδipöpi,헢룶뇈횵랴펳ퟔ컒움뚨볛룱뫍뿍맛움뚨볛룱횮볤뗄닮뻠.죴ri(p)>1,뇭쪾닺욷i뗄볛룱쿠뛔뇣틋;죴ri(p)<1,뇭쪾닺욷i뗄볛룱쿠뛔맽룟.뛔쿻럑헟살쮵,쯻쏇룼풸틢싲볛룱쿠뛔뇣틋뗄짌욷,틲듋,ri(p)풽듳,풽컼틽쿻럑헟.뚨틥di(p)=max{rj(p)ûjꇊI}-ri,돆캪뻠샫몯쫽.헢뇭쪾짺닺헟i뗄ri(p)폫ퟮ컼틽쿻럑헟뗄쓇룶짺닺헟횮볤뗄닮뻠.뛔쎿룶짺닺헟살쮵,di(p)풽킡풽뫃.di(p)풽킡,짺닺헟i뗄닺욷뗄탨쟳쿠뛔살쮵뻍풽듳.뛸짺닺헟붵볛뗄쒿뗄뻍쫇튪복킡ퟔ벺뗄di(p).쫗쿈,틽죫튻룶훸쏻Perron2Frobenius뚨샭.뚨샭1ꆡ짨A쫇럇뢺늻뿉풼랽헳,퓲듦퓚A뗄쳘헷횵Κ폫쿠펦뗄쳘헷쿲솿x싺ퟣ:(1)Ax=Κx,Κ>0,x>0,쟒뛔A뗄웤쯼죎틢쳘헷횵Α,폐Κ>ûΑû.(2)Κ쫇A뗄떥쳘헷룹.(3)Π쿲솿a1ya2yanyy>0,폐min,,ꆭ,ꇜΚꇜma1ya2yanyax,,ꆭ,.y1y2yny1y2ynΞ쫕룥죕웚:2000206228ퟷ헟볲뷩:췴볒뿛(싛19772),쓐,몺ퟥ,붭쯕뫩죋,훐맺뿆벼듳톧쫽톧쾵쮶쪿,퇐뺿랽쿲:퓋돯톧폫뿘훆.' 1995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.
62펦폃쫽톧2001볇헢룶쳘헷횵캪Κmax,돆캪A뗄훷쳘헷횵,뛔펦뗄쳘헷쿲솿돆캪훷쳘헷쿲솿.평폚Κmax쫇떥쳘헷룹,틲듋쳘헷ퟓ뿕볤M={xûAx=Κmaxx}쫇튻캬뗄,떱x캪훷쳘헷쿲솿쪱,틗횪폐:Κmax=a1xa2x==ꆭanx=.x1x2xn틽샭1ꆡ짨A쫇럇뢺늻뿉풼뻘헳,Κmax쫇A뗄훷쳘헷횵,죴x쫇A뗄튻룶쳘헷쿲솿쟒x>0,퓲x튻뚨쫇훷쳘헷쿲솿.횤ꆡ짨x뛔펦뗄쳘헷횵캪Κ,벴Ax=Κx.평x>0횪Κ쫇쪵쫽,쯹틔ΚꇜΚmax.Κ=a1xöx1=a2xöx2=ꆭ=anxöxn=max{a1xöx1,a2xöx2,ꆭ,anxöxn},평뚨샭1,Κmaxꇜmax{a1xöx1,a2xöx2,ꆭ,anxöxn}=Κ,쯹틔Κ=Κmax.틽샭2ꆡΚmax,Κδmax럖뇰쫇A,Aδ뗄훷쳘헷횵,퓲Κmax뫍Κδmax뛔펦쿠춬뗄헽쳘헷쿲솿,쟒Κδmax=(Κmax-1)ö(n-1)ꇝ1.횤ꆡ짨Κmax뛔펦뗄쳘헷쿲솿캪x,쟒짨x>0,벴Ax=Κmaxx,쯹틔Aδx=\[A-I]xö(n-1)=(Ax-x)ö(n-1)=(Κmax-1)xö(n-1).쯹틔x튲쫇Aδ뗄쳘헷쿲솿.죴x캪Aδ뗄헽훷쳘헷쿲솿,춬샭뿉횤x튲쫇A뗄훷쳘헷쿲솿.틲캪x>0,평틽샭1,퓲x쫇Aδ뗄훷쳘헷쿲솿.쯹틔Aδx=(Κmax-1)xö(n-1)=Κδmaxx,쯹틔Κδmax=(Κmax-1)ö(n-1)폖평Κmax>n(닎볻\[2\].p68),쯹틔Κδmaxꇝ1.뚨샭2ꆡ짨A쫇럇뢺뻘헳,aii=1,Aδ=(A-I)ö(n-1),Κδmax쫇A뗄훷쳘헷횵.죴x>0,쟒늻쫇Aδ뗄훷쳘헷쿲솿.퓲maδ1xaδδ2xnxin,,ꆭa,<Κδaδ1xaδ2xmax<max,,ꆭaδnx,x1x2xnx1x2xn횤ꆡ랴횤램.짨듦퓚θx>0쟒늻캪Aδ뗄훷쳘헷쿲솿,쟒Κδmaxꇜaδixθ믲δΚδixθmaxꇝa,ΠiꇊI돉xixi솢.쇮δixθxθJ=iꇊIûΚδamax=,퓲JꇙÁ,JꇙI.늻럁짨ΚδmaxꇜaδxθiꇊI돉솢,쯹xθ,Π틔ϖk,Κδaδkxθθxθi,ꆡꆡiꇙkmax<(럱퓲xθ캪쳘헷쿲솿).쇮0<∆aδkx<xθkΚδ-xθk,xζi=,퓲뗃쿲솿xζ=(xζ1,maxxθk+∆,i=kδxθ+aδik∆aixθxζ2,ꆭ,xζn)>0.떱i=k쪱,퓲aδkxζaδkxθxζ=ꇝΚδθ>kxθk+∆>Κδmax;떱iꇙk쪱,퓲aυixζaδixζ=iixθi벴aδixζxζ>Κδmax,ΠiꇊI.헢폫뚨샭1뷡싛(3)쎬뛜.i볙짨퇐뺿뗄슢뛏뺺헹탐튵쎻폐짺닺헟뗄뷸죫뫍췋돶,벴I={1,2,ꆭ,n}.퓚튻뛎쪱웚쓚,룷짺닺헟뗄닺욷훊솿튲컞뷏듳룄뇤,벴뇈뷏뻘헳A쫇뫣뚨늻뇤뗄.퓚슢뛏뺺헹쫐뎡훐볛룱햽쫇훷튪뺺헹쫖뛎.쒳룶짺닺헟붵볛뗄쒿뗄뻍쫇튪삩듳ퟔ벺닺욷뗄쫐뎡햼폐솿,떫듋짺닺헟뗄붵볛탐캪뇘좻떼훂웤쯼뎧짌튲럗럗붵볛,룷짺닺헟훆뚨뗄볛룱뚼쾣췻ퟔ벺뗄뻠샫몯쫽di(p)풽킡풽뫃.뚨틥뻹뫢볛룱Pθi,죧맻Pθ=(Pθ1,Pθ2,ꆭ,Pθn)싺ퟣ:di(θp1,pθ2,ꆭ,pθn)ꇜdi(θp1,ꆭ,pθi-1,pi,pθi+1,ꆭ,pθn),ΠiꇊI,Πpi>0뻹돉솢,퓲돆Pθ캪뻹뫢볛룱.뚨샭3ꆡ슢뛏뺺헹쫐뎡뗄뻹뫢볛룱Pθ쫇A뗄훷쳘헷쿲솿,랴횮튲돉솢.횤ꆡ짨뻹뫢볛룱Pθ늻쫇A뗄훷쳘헷쿲솿,평틽샭2,Pθ튲늻쫇Aδ뗄훷쳘헷쿲솿.평뚨샭2,δδδδϖakpθ1pθa2pθnpθζk,st.,pθ<maaxkpθ1pθ,ꆭa,2pθ,퓲dk(pθ)=max{rjûjꇊI}-rk>0.뚨틥pi=npθδδδkö2,i=k,짨alpθ1pθδδδa2pθθ퓲Pnpθalpθakpθ=(pζ1,pζ2,ꆭ,pζn).볙,pθ=maaxplpθk,iꇙθ,ꆭa,.퓲dk(pθ)=k,1p2pθnpθ-lpθ(lkδδδδδꇙaθδδjpζkpζjꇊa2akpθakpθk).dk(pζ)=maxI-malpaxpθjpζ=-kpθ,lpθkpθ.(1)죴aθlpθ2akp>ζk(p)=pθ,퓲dkpθlk' 1995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.
뗚2웚췴볒뿛뗈:폃AHP랽램럖컶슢뛏뺺헹쫐뎡63δδδδalpθ2akpθ2akpθζdk(pθ),pθ-lpθ<(2)죴alpθpθꇜlpθ,퓲dk(p)=0<dk(θp),벴폐ζdk(p)<dk(pθ).헢폫pθ쫇kk뻹뫢볛룱쎬뛜.쯹틔pθ튻뚨쫇훷쳘헷쿲솿.δδ랴횮θδanpθ,죴θp쫇훷쳘헷쿲솿.퓲a1pθa2ppθ==1pθ=ꆭ2pθ=Κδmax,쯹틔,di(θp)=nδδmajpθaipθaxpθjꇊI-jpθ=Κδmax-Κδmax=0<dθi(p1,ꆭ,pi,ꆭ,pθn).벴훷쳘헷쿲솿쫇뻹뫢볛룱.i뚨샭3횸돶,슢뛏뺺헹쫐뎡뗄뻹뫢볛룱쫇Aδ뗄훷쳘헷쿲솿.짨훷쳘헷횵Κδmax뛔펦뗄쳘헷ퟓ뿕볤M={PûAδP=Κδmaxp},M쫇튻캬쿟탔ퟓ뿕볤.짨M훐뗄떥캻쿲솿e,Aδe=Κδmaxe,틲듋뿉짨뻹뫢볛룱pθ=ae,a>0돆a캪볛룱쾵쫽,쓇쎴,a좡뫎횵닅쪹짺닺헟듯떽샻죳벫듳뮯쓘?쿂쏦살럖컶짺닺헟뗄샻죳몯쫽.짨P=(p1,p2,ꆭ,pn)캪볛룱쿲솿,q=(q1,q2,ꆭ,qn)캪닺솿.wi(p)뇭쪾닺욷i뗄탨쟳몯쫽,Ci(qi)쫇짺닺헟i뗄돉놾몯쫽.틲듋짺닺헟i뗄샻죳몯쫽뇭쪾죧쿂:Πi(p,qi)=piwi(p)=ci(qi),qi=wi(p).쓇쎴퓚뻹뫢볛룱쳵볾쿂,짏쪽뿉킴돉:Πi(a,qi)=aewi(ae)=ci(qi),qi=wi(ae).짺닺헟i튪쪵쿖샻죳벫듳뮯,뻍쫇튪뻶뚨볛룱쾵쫽a,듓뛸뻶뚨닺솿qi,틲듋쯼뛔펦튻룶맦뮮컊쳢:maxΠi(a,qi)=aewi(ae)-ci(qi),=wi(ae).볇웤ퟮ폅뷢캪(ai,qi),퓲ai뇭쪾짺닺헟i쯹뚨뗄볛룱쾵쫽,돆캪짺닺헟i뗄ퟮ폅볛룱쾵쫽.떫펦룃힢틢떽,늻춬짺닺헟뗄ퟮ폅볛룱쾵쫽쫇늻튻뚨쿠춬뗄.볇a=min(aiûiꇊI},짨a=ak,kꇊI,퓲짺닺헟k퓚쿠뛔뷏뗍뗄뻹뫢볛룱쮮욽짏뿉틔듯떽쯼뗄샻죳벫듳뮯,뇭쪾짺닺헟k퓚볛룱뺺헹짏뒦폚쿠뛔폐샻뗄캻훃,폐잿듳뗄뺺헹쪵솦.짺닺헟k췹췹쫇볛룱햽뗄쪼ퟷ?헟,쯼붫볛룱붵떽ퟮ뗍쮮욽,좴잡잡쪵쿖쯼뗄샻죳벫듳뮯.쇮pθ=ae,퓲pθ뻍쫇슢뛏뺺헹쫐뎡뗄뻹뫢볛룱.닎뾼컄쿗\[1\ꆡ]쯎돐쿈.쿖듺컷랽뺭볃톧(캢맛뺭볃톧)[M\].짏몣:뢴떩듳톧돶냦짧,1999.\[2\ꆡ]뫮뚨?.닣듎럖컶[M\].낲믕:낲믕뿆톧벼쫵돶냦짧,1990.\[3\ꆡ]룊뮪쏹뗈틫,죸뛻톷.뺭볃럖컶믹뒡[M\].놱뺩:놱뺩뺭볃톧풺돶냦짧,1990.\[4\ꆡ]엋벪톫,헅쮳쏷.뺭볃뻹뫢뗄쫽톧풭샭[M\].벪쇖:벪쇖듳톧돶냦짧,1997.\[5\ꆡ][J\].JournaloftheOperationsResearchSocietyofJapan.,1999,6:219ꆫ,LINChuan(USTCMathematicsDepartment,AnHuiHeFei230026,China)Abstract:Inthispaper,:Pairwisecompairisonmatrix;Self2evaluationprice;Non2self2evaluationprice;Thedistantfunction;Equilibriumprice' 1995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.
