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湖南大学硕士学位论文最优化方法及其在供应链库存管理中的应用姓名：谢水连申请学位级别：硕士专业：运筹学与控制论指导教师：李董辉20070427 :4?9%jxL$w|;[ x ! $ Y ! YI3 !9k I3$hvJz9Z v .BFGSI3#z Pp 3 ! Y4L4 NewtonI3Gg.TI3 Y V&D#ID.^-, 3 4z?g 4Æ, Newton3 Y #ID.zQ JBFGSI3 :gX_Z$,Li-Fukushima(2001)(3oBFGSI3{CQ U{,XgQg^GA BFGS(MBFGS)I3,6D Q$hp ^^ ,C Wolfe-PowellA ArmijoA? \&DFNÆ,I3( Pp?g 42L Yd Y #ID.$ dZ Gg Y Li-Fukushima(2001)Xg MBFGSI3Z = #ID. ~D ,$g ^^ ,MBFGSI3 PpX^g/t 42LÆN =MS #I .$nd O +Kt#v ! AZ, }+4 #v ! A 7 hvJ ,Ax T(u(N0hvu)–C/Z?(,0u)–Y*u qKt#v !. y x Rhvu–T(u–C/uGW #v ! A.$ [g_ h !93$ [Z v ,Y $Æ_ hvJz9Zj Æ9 4 v !),k6z 4p } . ~( +hvu+T(u +Y*uoZ qKthvJ #v A{CQ kY ._Dg_ HPO}, _sd 3t{HP,+_T(u-+_hvu{5Æ,hvJvj 6 k _hvJ #vxB z 2 A. f$Æa6Hi 4!a6Hi / 4 !a6Hiq^6 8T C)O {^8TZhvJ{_Z 5a6.Pn a J8 [ MBFGS?; v$ u L m2b-II S n AbstractThisthesisstudiesalgorithmsforoptimizationandtheapplicationsofopti-mizationtheoryandalgorithmsinsupplychainmanagement.Amongthequasi-Newtonmethodsforsolvingunconstrainedoptimizationproblems,theBFGSmethodisregardedasoneofthemostefficientmethods.Itpossessesgoodglobalandsuperlinearconvergenceproperties.Nevertheless,whentheobjectivefunctionisnon-convex,thequasi-Newtonmethodmaynotbeglob-allyconvergent.Asaremedy,Li-Fukushima(2001)proposedamodifiedBFGS(MBFGS)methodbymakingsomepropermodificationtothestandardBFGSmethod.Underweakconditions,theyprovedthatthemethodwithWolfe-PowelltypelinesearchorArmijotypelinesearchisstillgloballyconvergentevenwhentheobjectivefunctionisnotconvex.Onepurposeofthisthesisistoinvestigatetheconvergenceofmatricesse-quencegeneratedbytheMBFGSmethod.Weprovethat,undercertaincondi-tions,thesequenceconvergesiftheobjectivefunctionisastrictlyconvexquadraticfunction.Traditionallymostmulti-stageinventorymodelsinmanysupplychainwerebasedonthedownstreamstage.Inotherwords,themembersinsupplychainarethemanufacturer(supplieroftheproduct),thewholesalerandtheretailer.Intheupstreamstage,peoplejustconsidertheproblemlikechoiceofsuppliers.Fewerworkisconcernedwithinventorymodelinvolvingvendor,manufacturerandretailer.Anotherpurposeofthisthesisistoapplyoptimizationtheoryandalgorithmstosupplychainandlogisticsmanagementtomaximizetheprofit.Weproposeathreestagesupplychaininventorymodelinvolvingmulti-vendor,multi-manufacturerandmulti-retailer.Wegiveanoptimalcoordinationstrategyofhowtocoordinatethesupplychainsuchthatthewholecostofinventoryofsupplychainisminimized.Weconsiderthreecoordinationmechanisms,i.e.,equalcycletime,integer-multiplierateachstageandintegerpowersoftwomultipliersateachfirm.KeyWords:unconstrainedoptimization;MBFGSmethod;convergenceofmatrices;supplychain;inventory;coordinationmechanismIII S n Z%$ Z ℄ v: $aC_ O[ k $a$ M D{CY OU Z hQ ZV5Qk3ht hn $k b 8P_a4=Zj 044k8 Z 0 ($ Y g_l# _a =Z j$ Zk \9 3 $aoYn $ 3g $a_ 9B e6 e Z ℄ 1? D1F$P k 9oYQpP3 x ℄ P k bnP3 ℄6- P x?}48oEk Ou^ s Kk #L s $a(W }P k‘$P k Y?4?C h.k x4 #{CXN kC |u Mu4s ÆOT$% v ..$P k $P k / 1 $ p $(W. 2 (M$kv’v9) | X ) 9B e6 e B e6 eI S n 11#} 1.1; ^ C1.1.18 [_SHH !93$ Fj e$e N a, ug xm z9 PY Z p~8 v .$ Me Z, !& [g^ o,A 3 Bp },j $ Nz9Zj }%# b Z$ j Æ9 R m R j }I,9 ,B y;j IIhvJv‘m #vO #vajÆWk _hvJ#vB yÆ. s F,O ! $‘+ C9 493Z5 9 493.sÆ, ! Y 4P Aj minf(x),x∈D⊆Rn,(1.1)8Z, 4f o$Rnv L 4, ~Y 4fz Y(1.1) 3 4. D=RnÆ, ~Yv1 Yz 3 ! Y,YKzminx∈Rnf(x).(1.2) 3 ! YQC 3P p.}zPp93AMq5, O 93 17$ ^ I31.1 !93 7$ ^ 0:ag p g_f sIx(0),Sk:=0 1:j x(k)t{( psI ℄N^^,‘N 2:\ g_Utx(k) GAOδ(k) 3: p g_d sIx(k+1)=x(k)+δ(k),Sk:=k+1,j1.zQ v I3 Y #ID,aQC &DFNE0,sÆ’v ^ z:x(k+1)=x(k)+αkd(k),(1.3)8Zd(k)zFN9-,Rssαk &DFN\ .Q &DFN93 N^9 ,A \&DFN ? \&DFN. \&DFN\ Rαk j g{ ! Y p minα0f(x(k)+αd(k)),ϕ(α).(1.4) \&DFN ^^mF, \&DFN\ Rαkt{ ∇f(x(k)+αkd(k))Td(k)=0.(1.5)–1– :4?9%jxL$w|;[ x \&DFN$Pp Y(1.4) 2L Æ,IIO },V5,( 5?} 44 44z O 4 B \&DFN93.Ok Cd+C ? \&DFN.sÆ,\ RαkÆ,OH 3 4f$x(k)+αkd(k) Lf(x(k)+αkd(k))hf(x(k)) g \# bA .Q ? \&DFN9 N^ AArmijoA&DFN Wolfe-PowellA&DFN.ArmijoA&DFN,ARαk0t{j Æ f(x(k)+αkd(k))≤f(x(k))+σαk∇f(x(k))Td(k),(1.6)8Zσ∈(0,1)za Q4,Rαk U=Æ{1,ρ,ρ2,ρ3,···}Z v ZD }L,ρ∈(0,1)zga Q4.Wolfe-PowellA&DFN,Aa Q4σ1,σ2t{0σ112,σ1σ21,bÆt{ N_Æ (f(x(k)+αkd(k))≤f(x(k))+σ1αk∇f(x(k))Td(k),f(x(k)+αkd(k))Td(k)≥σ2∇f(x(k))Td(k).(1.7) (1.3) F,\ I31.12Z GAOδ(k),hQ\ RssαkGn,&H\ FN9-d(k),:d _I3 ?.b I3\ d(k) 9 b,’vI3 #ID9k 4L4 d } Mr. ~0tzg Pp 3 Y(1.2) C^I3._ tz Newton3.1.1.2a J_SN;1 H b3: Y g^I3, f 3 4PPCauchy 1874 Xg.Q kRW#9- z b9- 2!I3.s93 vkOy V , #IH#hu.2Newton3} 3 4f(x)/tGN v,B f(x)$x(k)m /tTalor. z A 4,6 :_/t A 4 2 MSV%| 3 4 2 .8 k z x(k+1)=x(k)−[∇2f(x(k))]−1∇f(x(k)).(1.8)( A /t 4,Newton3gA z p. ( g ?g/t 4,6 D %t P p.3 Newton3–2– S n Newton3 $Newton37jvN .v ; Newton3Zg x℄ B Q 3 4f(x) Hessian =Xh Sh ,u- /t#ID. TI3HII 3 4 Hessian =,IIO},-G ∇2f(x(k))A Æ, DI3N 9- f$x(k)m b9-.V5 , ∇2f(x(k)):rÆ,Newton9- v$,u-|*I3 Y #ID. Newton3 7$) ∇2f(x(k)) g_|D =BkV [Newton3Z Hessian =.8I3 7$j I31.2 &DFNA NewtonI3 0:Uf