一个具有可调变权能力的变权向量

271Vol.27No.1ControlandDecision20121Jan.2012:1001-0920(2012)01-0082-051;2,1(1.7100712.725000):,;,,.,,,.::O159:AAvariableweightsvectorwithadjustablecapabilitytochangeweightsCHENGBo1;2,LIUSan-yang1(1.SchoolofScienceXidianUniversityXi’an710071China2.DepartmentofMathematicsAnkangUniversity,[e-mail])Abstract:Avariableweightvectorwithparametersisconstructedbyusingitsdefinition.Atheoremfortherelationshipofavariableweightvectoranditsstatevariableweightisproved.Byintroducingaconceptofrelativeadjustmentdegree,thecapabilityofavariableweightvectorismeasuredtochangeweights.Therelativeadjustmentdegreeofthevariableweightvectoriscalculated,anditscapabilitytochangweightsisanalyzed.Finally,anexampleissolvedbyusingthevariableweightvector.Theresultshowsthatnotonlythevariableweightvectorhasstrongcapabilitytoadjustweights,butalsoitscapabilitytoadjustweightschangeswhenthevalueoftheparameterisadjusted.Keywords:multi-objectivedecision-makingvariableweightssynthesisvariableweightsvectorstatevariableweightvectorrelativeadjustmentdegree1,,.,.,,[1].,[1];[2-4],,,.,[5-8].[9-10],,.,[11-13].,.[14],.,,;,,.2[2],[5-6].1[0;1]mmRm,wj:[0;1]m,x=(x1;x2;¢¢¢;xm)2[0;1]m.w(x)=(w1(x);w2(x);¢¢¢;wm(x))::2010-08-30:2010-12-27.:(60974082).:(1971¡),,,,(1959¡),,,,.1:83(W1):mXj=1wj(x)=1;(W2):wj(x1;x2;¢¢¢;xm)(j=1;2;¢¢¢;m).w(x)x,.maxx2[0;1]m³mXj=1xjwj(x)´:w(x):(W3):wj(x1;x2;¢¢¢;xm)xj(j=1;2;¢¢¢;m),w(x).w(x):(W30):wj(x1;x2;¢¢¢;xm)xj(j=1;2;¢¢¢;m),w(x).2sj:[0;1]m,s(x)=(s1(x);s2(x);¢¢¢;sm(x)):(S1)x=(x1;x2;¢¢¢;xm)2[0;1]m,xixj,si(x)6sj(x);(S2)sj(x1;x2;¢¢¢;xm)(j=1;2;¢¢¢;m);(S3)w=(w1;w2;¢¢¢;wm),w(x)1(W1),(W2)(W3),w(x)=(w1s1(x);¢¢¢;wmsm(x))mXj=1(wjsj(x)):(1)s(x),(1)ws(x)Hadamard.(S1):(S10)x2[0;1]m,xixj,si(x)sj(x);(S3)(W3)(W30),s(x).,(),(),[2].,,Hadamard?.1w=(w1;w2;¢¢¢;wm)wj0(j=1;2;¢¢¢;m);x2[0;1]m;w(x)=(w1(x);w2(x);¢¢¢;wm(x)).1)w(x),s(x)w(x)=(w1s1(x);¢¢¢;wmsm(x))mXj=1(wjsj(x)):x=(x1;x2;¢¢¢;xm)2[0;1]m,xixj,wi(x)wi6wj(x)wj;2)w(x),s(x)w(x)=(w1s1(x);¢¢¢;wmsm(x))mXj=1(wjsj(x)):x=(x1;x2;¢¢¢;xm)2[0;1]m,xi6xj,wi(x)wi6wj(x)wj.1).w(x),x2[0;1]mwj(x)0mXj=1wj(x)=1.wj0,sj(x)=wj(x).³wjmXi=1wi(x)wi´;(2)s(x)=(s1(x);s2(x);¢¢¢;sm(x)).s(x).,wj0;wj(x)0,(2)06sj(x)61.,wj(x),sj(x),s(x)(S2).,w(x)=(w1s1(x);¢¢¢;wmsm(x))mXj=1(wjsj(x));s(x)(S3).x2[0;1]m,xixj,wi(x)=wi6wj(x)=wj,si(x)¡sj(x)=wi(x)wi¡wj(x)wj60;s(x)(S1).2,s(x)..[10]1.2)1).21,,,1wj0(j=1;2;¢¢¢;m).,.3x2[0;1]m;wwj0(j=1;2;¢¢¢;m),w(x),vj(w(x))=wj(x)¡wjwjw(x)xwj,8427¹v(w(x))=vuutmXj=1(vj(w(x)))2w(x)x.vj(w(x))wj(x)wj.vj(w(x)),xj;vj(w(x)),xj.w(x)w.,,,,.3,.,,[14].w=(w1;w2;¢¢¢;wm),w(x)=(w1(x);w2(x);¢¢¢;wm(x)),wj(x)=(1+®(¹x¡xj))wj:(3):j=1;2;¢¢¢;m;x2[0;1]m;¹x=mXj=1(xjwj);®1minx2[0;1]mmin16j6m(xj¡¹x)6®61maxx2[0;1]mmax16j6m(xj¡¹x):(4)2(3)w(x).®0,w(x);®0,w(x).w(x)1.0®61maxx2[0;1]mmax16j6m(xj¡¹x);x2[0;1]m,¹x¡xj0wj(x)=(1+®(¹x¡xj))wj0;¹x¡xj0,wj(x)=(1+®(¹x¡xj))wj³1+¹x¡xjmaxx2[0;1]mmax16j6m(xj¡¹x)´wj³1+¹x¡xjxj¡¹x´wj=0:mXj=1wj(x)=mXj=1(1+®(¹x¡xj))wj=mXj=1wj+®³¹xmXj=1wj¡mXj=1xjwj´=1;w(x)1.@wj(x)=@xj=®(wj¡1)0,wj(x)xj,w(x).w(x)1.,wj(x)wj=1+®Xi6=jxiwi+®(wj¡1)xjxj,11).sj(x)=1+®(¹x¡xj)m(1+®¹x)¡mXj=1xj;j=1;2;¢¢¢;m;(5)1,s(x)=(s1(x);s2(x);¢¢¢;sm(x))w(x).1minx2[0;1]mmin16j6m(xj¡¹x)6®0,w(x)1,(5),.22,®0,w(x);®0,w(x);®=0,w(x)=w.,w(x)®,,.,(3)w(x)xjvj(w(x))=®(¹x¡xj),¹v(w(x))=j®jvuutmXj=1(¹x¡xj)2;xj®j.®x,.,x:x(x1=x2=¢¢¢=xm),¹v(w(x))=0,x,w(x)=w;x,¹v(w(x)),,.,x2[0;1]m,w(x),j®j®.®=1minx2[0;1]mmin16j6m(xj¡¹x),w(x);®=1maxx2[0;1]mmax16j6m(xj¡¹x),w(x).4[8](3).1:85,(w1),(w2)(w3)3,A,B,C,D,E5,12.10.360.310.332A0.2140.1660.184B0.2060.2200.182C0.1950.1920.220D0.1810.1950.185E0.1750.1930.201,M0(x)=3Xj=1(xjwj),M0(A)=0:1892,M0(B)=0:2024,M0(C)=0:2023,M0(D)=0:1867,M0(E)=0:1892.BÂCÂA»EÂD.[8]B»CÂAÂEÂD.(3),M(x)=3Xj=1(xjwj(x)):(6),(4)¡70:6216®640:355.®=1,(3),3.3A0.3510.3170.332B0.3590.3040.337C0.3630.3130.324D0.3620.3070.331E0.3650.3090.3264A0.2710.3820.347B0.3470.2560.397C0.3860.3420.272D0.3800.2840.336E0.4110.2980.291(6)M(A)=0:1888,M(B)=0:2022,M(C)=0:2022,M(D)=0:1866,M(E)=0:1890.C»BÂEÂAÂD.,,..,¹v(w(A))=0:0344,¹v(w(B))=0:0272,¹v(w(C))=0:0217,¹v(w(D))=0:0102,¹v(w(E))=0:0189.A,A5.,Aw1,¡2:5%.®=10,(3)4.(6)M(A)=0:1852,M(B)=0:2000,M(C)=0:2008,M(D)=0:1863,M(E)=0:1879.CÂBÂEÂDÂA.,,.¹v(w(A))=0:344,¹v(w(B))=0:272,¹v(w(C))=0:217,¹v(w(D))=0:102,¹v(w(E))=0:189.Aw1,¡25%.,,,,,.5,.,,,.,,.,,.(References)[1].[M].:,1985:47-59.(WangPZ.Shadowoffuzzysetsandrandomsets[M].Beijing:BeijingNormalUniversityPress,1985:47-59.)[2].()[J].,1995,9(3):1-9.(LiHX.Factorspacesandmathematicalframeofknowledgerepresentation()[J].FuzzySystemsandMathematics,1995,9(3):1-9.)[3].()[J].,1996,10(2):12-19.(LiHX.Factorspacesandmathematicalframeof8627knowledgerepresentation()[J].FuzzySystemsandMathematics,1996,10(2):12-19.)[4].()[J].,1996,10(4):110-118.(LiHX.Factorspacesandmathematicalframeofknowledgerepresentation()[J].FuzzySystemsandMathematics,1996,10(4):110-118.)[5],.[J].,1999,19(7):116-118.(ZhuYZ,LiHX.Axiomaticsystemofstatevariableweightsandconstructionofbalancefunctions[J].SystemsEngineering–TheoryandPractice,1999,19(7):116-118.)[6]