뗚20뻭뗚5웚ꆡꆡꆡꆡꆡꆡ쾵ꆡ춳ꆡ릤ꆡ돌ꆡ톧ꆡ놨2005쓪10퓂ꆡꆡꆡꆡꆡJOURNALOFSYSTEMSENGINEERINGꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡힰ킶믬뫏뎵솾슷뺶컊쳢뗄쒣쓢췋믰쯣램퇐뺿ꋙ색쎯쿩(놱뺩붻춨듳톧붻춨퓋쫤톧풺,놱뺩100044)햪튪:쳡돶쇋룼뻟튻냣탔뗄ힰ킶믬뫏뎵솾슷뺶컊쳢,붨솢쇋룃컊쳢뗄믹폚횱맛쏨쫶뗄쫽톧쒣탍.춨맽짨볆튻훖탂뗄뷢뗄뇭쪾랽램릹퓬쇋쟳뷢룃컊쳢뗄쒣쓢췋믰쯣램,늢뷸탐쇋쪵퇩볆쯣.볆쯣뷡맻뇭쏷,폃짨볆뗄쒣쓢췋믰쯣램쟳뷢ힰ킶믬뫏뎵솾슷뺶컊쳢,늻뷶뿉틔좡뗃뫜뫃뗄볆쯣뷡맻,뛸쟒볆쯣킧싊뷏룟,쫕솲쯙뛈뷏뿬,볆쯣뷡맻튲뷏컈뚨.춨맽뛔쮫쿲엤쯍닟싔폫떥쿲엤쯍닟싔볆쯣뷡맻뗄뇈뷏,쮵쏷쇋닉폃쮫쿲엤쯍닟싔쟳뷢ힰ킶믬뫏뎵솾슷뺶컊쳢뛔폚엤쯍웳튵뷚쪡엤쯍뎵솾ꆢ복짙엤쯍샯돌,듓뛸붵뗍엤쯍돉놾ꆢ쳡룟뺭볃킧틦뗄훘튪틢틥.맘볼듊:ힰ킶믬뫏뎵솾슷뺶컊쳢;쒣쓢췋믰쯣램;폅뮯훐춼럖샠뫅:U116ꆡꆡꆡ컄쿗뇪쪶싫:Aꆡꆡꆡ컄헂뇠뫅:1000-5781(2005)05-0485-07StudyonsimulatedannealingalgorithmforvehicleroutingproblemwithbackhaulsLANGMao2xiang(SchoolofTrafficandTransportation,BeijingJiaotongUniversity,Beijing100044,China)Abstract:,-directiondistributiontacticsandthoseoftheone-directiondistributiontactics,thispaperalsoillustratestheadvantagesbyusingtwo-directiondistributiontacticstosolvethevehicleroutingproblemwithbackhaulsforphysicaldistributionenterprisestosavingdistributionvehicles,:vehicleroutingproblemwithbackhauls;simulatedannealingalgorithm;optimal0ꆡ틽ꆡ퇔뚼뿉틔돩쿳캪뎵솾슷뺶컊쳢.틲듋,퇐뺿뎵솾슷뺶컊쳢뻟폐훘튪뗄샭싛뫍쿖쪵틢틥.뎵솾슷뺶컊쳢(vehicleroutingproblem,VRP)캪쇋쪹컊쳢틗폚샭뷢,놾컄틔컯쇷엤쯍캪놳쫇떱잰퓋돯톧폫ퟩ뫏폅뮯쇬폲뗄잰퇘폫죈뗣컊뺰퇐뺿뎵솾슷뺶컊쳢,벴퇐뺿컯쇷엤쯍훐뗄뎵솾쳢.퓚쿖쪵짺닺뫍짺믮훐,컯쇷엤쯍뎵솾뗷뛈컊슷뺶컊쳢.쳢ꆢ폊헾춶뗝컊쳢ꆢ릫릲웻뎵뗷뛈컊쳢ꆢ뗧솦뗷뛈쿖폐컄쿗퇐뺿뗄뎵솾슷뺶컊쳢뛠캪떥쿲뎵솾컊쳢ꆢ맜뗀워짨컊쳢ꆢ볆쯣믺췸싧췘웋짨볆컊쳢뗈슷뺶컊쳢,냼삨듓즢믵탍컯쇷훐탄쿲뿍뮧쯍믵뗄뒿ꋙ쫕룥죕웚:2003-07-24;탞뚩죕웚:2004-01-14.
ꆪ486ꆪ쾵ꆡ춳ꆡ릤ꆡ돌ꆡ톧ꆡ놨ꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡ뗚20뻭쯍믵컊쳢뫍붫믵컯듓뿍뮧좡떽벯믵탍컯쇷훐탄뗄뗄볆쯣뷡맻,뛸쟒쯣램뗄볆쯣킧싊뷏룟,쫕솲쯙뛈뒿좡믵컊쳢.돽쇋즢믵탍뫍벯믵탍컯쇷훐탄췢,쿖뷏뿬,볆쯣뷡맻튲뷏컈뚨.춨맽뛔쮫쿲엤쯍닟싔폫쪵훐뮹폐탭뛠벯즢탍뗄컯쇷훐탄,죧쳺슷믵뎡ꆢ릫떥쿲엤쯍닟싔뗄볆쯣뷡맻뗄뇈뷏,쮵쏷닉폃쮫쿲엤슷훐햾벰룛뿚싫춷뗈.뛔폚헢샠벯즢탍컯쇷훐쯍닟싔쟳뷢ힰ킶믬뫏뎵솾슷뺶컊쳢뛔폚엤쯍웳튵탄,퓚훆뚨엤쯍랽낸쪱,볈튪뾼싇붫뿍뮧탨튪뗄믵뷚쪡엤쯍뎵솾ꆢ복짙엤쯍샯돌,듓뛸붵뗍엤쯍돉놾ꆢ컯듓컯쇷훐탄쯍떽룷룶뿍뮧,폖튪뾼싇붫뿍뮧짺닺쳡룟뺭볃킧틦뗄훘튪틢틥.믲릩펦뗄믵컯듓뿍뮧좡떽컯쇷훐탄,놾컄돆짏쫶컊쳢캪ힰ킶믬뫏뎵솾슷뺶컊쳢.뛔폚짏쫶컊쳢,뿉틔1ꆡힰ킶믬뫏뎵솾슷뺶컊쳢뗄쫽톧쒣탍붫쯍믵뫍좡믵럖뇰뾼싇,벴춨맽쟳뷢솽룶떥쿲뎵솾슷뺶컊쳢,럖뇰좷뚨쯍믵뗄엤쯍슷쿟뫍좡믵뗄엤쯍ힰ킶믬뫏뎵솾슷뺶컊쳢뿉틔쏨쫶캪:폃뛠첨슷쿟,놾컄붫짏쫶엤쯍닟싔돆캪떥쿲엤쯍닟싔.뛸엤쯍뎵솾붫뿍뮧탨튪뗄믵컯듓컯쇷훐탄쯍떽룷룶뷢뻶룃컊쳢뗄룼캪뿆톧뫏샭뗄랽램쫇붫쯍믵뫍좡뿍뮧,늢붫뿍뮧릩펦뗄믵컯듓룷룶뿍뮧좡떽컯쇷믵춳튻뾼싇,살뻶뚨믵컯뗄엤쯍슷뺶,벴퓚룷쳵엤훐탄,쎿룶뿍뮧뗄캻훃튻뚨,믵컯탨쟳솿뫍릩펦솿쯍슷뺶훐,뎵솾볈췪돉쯍믵죎컱,폖췪돉좡믵죎컱,튲튻뚨,쎿첨엤쯍뎵솾뗄퓘훘솿튻뚨,웤튻듎엤쯍돆짏쫶엤쯍닟싔캪쮫쿲엤쯍닟싔.뗄ퟮ듳탐쪻뻠샫튻뚨,튪쟳뫏샭낲업뎵솾엤쯍슷GoetschalckxꆢShangꆢThangiahꆢDuhamelꆢTothꆢ쿟,쪹쒿뇪몯쫽뗃떽폅뮯,늢싺ퟣ틔쿂풼쫸쳵볾:MosheiovꆢSwihartꆢNanryꆢWade뗈뚼퓸퇐뺿맽ힰ킶1)퓚엤쯍맽돌훐쎿첨엤쯍뎵솾뗄퓘믵솿늻뎬맽웤믬뫏뎵솾슷뺶컊쳢[1ꆫ10],늢좡뗃쇋뫜뛠퇐뺿돉ퟮ듳퓘훘솿;2)쎿쳵엤쯍슷뺶뗄뎤뛈늻뎬맽엤쯍맻.떫튻냣뚼믹폚죧쿂ힰ킶믬뫏뎵솾슷뺶컊쳢:퓚뎵솾튻듎엤쯍뗄ퟮ듳탐쪻뻠샫;3)쎿룶뿍뮧탨쟳쒳엤쯍쾵춳훐튻늿럖뿍뮧캪탨쟳뿍뮧,쇭튻늿럖뗄믵컯뇘탫쯍떽,릩펦뗄믵컯뇘탫좡ퟟ.뿍뮧캪릩펦뿍뮧.놾컄쳡돶쇋룼뻟튻냣탔뗄ힰ킶뛔폚ힰ킶믬뫏뎵솾슷뺶컊쳢,퓚엤쯍맽돌훐,믬뫏뎵솾슷뺶컊쳢:퓚쒳엤쯍쾵춳훐,볙짨쯹폐뗄엤쯍뎵솾뺭맽룷뿍뮧쪱,뿉틔쿈붫룃뿍뮧탨쟳뗄뿍뮧볈쫇릩펦뿍뮧,튲쫇탨쟳뿍뮧,죴쒳뿍뮧뷶릩믵컯킶쿂,좻뫳ힰ짏룃뿍뮧릩펦뗄믵컯,벴뎵솾뿉펦믵컯,뿉좡웤믵컯탨쟳솿캪0,죴쒳룶뿍뮧뷶탨퓚쯍믵뗄춬쪱췪돉좡믵죎컱.평폚믵컯뗄ힰ킶ퟷ쟳믵컯,퓲뿉좡웤믵컯릩펦솿캪0.내뿍뮧뛔믵컯튵쫇퓚엤쯍뎵솾뗄튻뛋뷸탐뗄,퓲엤쯍뎵솾퓚좡좡ꆢ쯍쪱볤뗄튪쟳,ힰ킶믬뫏뎵솾슷뺶컊쳢뿉럖캪쯍믵맽돌훐뷸탐믵컯떹ힰ헻샭쫇늻뿉뇜쏢뗄.캪컞쪱쿞컊쳢(뿍뮧뛔믵컯뗄좡ퟟ믲쯍떽뗄쪱볤튪쇋복짙엤쯍훐뗄닮듭늢복짙믵컯뗄떹ힰ릤ퟷ솿,쟳늻퇏룱,벴믵컯뿉퓚쒳쳬뗄죎뫎쪱볤쯍떽믲좡엤쯍뎵솾퓚헻룶좡쯍믵맽돌훐튪ퟱ톭ꆰ뫳떽쿈ힰꆱퟟ)뫍폐쪱쿞컊쳢(뿍뮧튪쟳붫탨쟳뗄믵컯퓚맦뚨뗄풭퓲:퓚컯쇷훐탄ힰ룷뿍뮧탨쟳뗄믵컯쪱,붫뫳뗄쪱볤뒰쓚쯍떽,붫릩펦뗄믵컯퓚맦뚨뗄쪱볤뒰떽뿍뮧탨쟳뗄믵컯ힰ퓚뎵솾뗄샯뛋,붫뗚튻룶뿍쓚좡ퟟ,튲돆캪폐쪱볤뒰컊쳢).놾컄훷튪퇐뺿컞뮧탨쟳뗄믵컯ힰ퓚뎵솾뗄ퟮ췢뛋;떽듯뗚튻룶뿍쪱쿞뗄ힰ킶믬뫏뎵솾슷뺶컊쳢.뮧뫳,붫룃뿍뮧탨쟳뗄믵컯킶쿂늢붻믵,좻뫳춨맽쒣쓢췋믰쯣램뗄돶쿖,캪쟳뷢뎵솾슷뺶컊쳢쳡떹ힰ헻샭ퟷ튵붫룃뿍뮧릩펦뗄믵컯ힰ퓚뎵솾뗄샯릩쇋탂뗄릤뻟.뷼쓪살,OsmanꆢLaporteꆢTeodorovicꆢ뛋,붫쿂튻룶뿍뮧탨쟳뗄믵컯ힰ퓚뎵솾뗄ퟮ췢뛋;닌퇓맢ꆢ쇵뫆뗈죋뚼퓸샻폃쒣쓢췋믰쯣램쟳뷢뎵솾틀듋샠췆,횱떽뎵솾췪돉좡쯍믵죎컱.뿉볻,뛔폚슷뺶컊쳢[11ꆫ15],늢좡뗃쇋뫜뛠돉맻.떫튻냣뷶쿞폚ힰ킶믬뫏뎵솾슷뺶컊쳢,닉폃쮫쿲엤쯍닟싔뇈닉뛔떥쿲뎵솾슷뺶컊쳢믲싺퓘뎵솾슷뺶컊쳢뗄쟳뷢,폃떥쿲엤쯍닟싔퓶볓쇋튻뚨뗄믵컯떹ힰ릤ퟷ솿,폃쒣쓢췋믰쯣램쟳뷢ힰ킶믬뫏뎵솾슷뺶컊쳢뗄퇐늢퓬돉뎵솾퓚뿍뮧뒦ퟷ튵춣쇴쪱볤뗄퇓뎤,헢뛔뺿뫜짙.놾컄춨맽짨볆튻훖탂뗄뷢뗄뇭쪾랽램릹퓬쟳뷢폐쪱쿞ힰ킶믬뫏뎵솾슷뺶컊쳢펰쿬뫜듳(ퟷ쇋쟳뷢ힰ킶믬뫏뎵솾슷뺶컊쳢뗄쒣쓢췋믰쯣램,늢헟붫쇭췢컄폨틔뾼싇),본폚놾컄훷튪퇐뺿컞쪱뷸탐쇋볆쯣.뷡맻뇭쏷,폃놾컄짨볆뗄쒣쓢췋믰쯣쿞뗄ힰ킶믬뫏뎵솾슷뺶컊쳢,뿉틔뫶싔틲믵컯떹램쟳뷢ힰ킶믬뫏뎵솾슷뺶컊쳢,늻뷶뿉틔좡뗃뫜뫃ힰ뫍뎵솾퓚뿍뮧뒦춣쇴쪱볤퇓뎤뛔컊쳢뗄펰쿬.
뗚5웚ꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡ색쎯쿩:ힰ킶믬뫏뎵솾슷뺶컊쳢뗄쒣쓢췋믰쯣램퇐뺿ꆪ487ꆪ짨컯쇷훐탄폐K첨엤쯍뎵솾,뎵솾k뗄퓘훘뗄,틲듋,튪놣횤퓚엤쯍맽돌훐뎵솾뗄퓘믵솿쪼훕솿캪Qk(k=1,2,ꆭ,K),웤튻듎엤쯍뗄ퟮ듳탐쪻늻뎬맽웤퓘훘솿,뇘탫싺ퟣ틔쿂쳵볾:ꋙ떱뎵솾뻠샫캪Dk,탨튪쿲L룶뿍뮧쯍믵뫍좡믵,뿍뮧i뗄듓컯쇷훐탄돶랢쪱,뎵짏ힰ퓘뗄쫇뿍뮧1ꆢ3ꆢ6탨탨쟳솿캪qiꆢ릩펦솿캪u(ii=1,2,ꆭ,L),뿍뮧i쟳뗄믵컯,헢쪱튪싺ퟣq1+q3+q6ꇜQ;ꋚ뎵솾떽j뗄퓋뻠캪d(iji,j=1,2,ꆭ,L),컯쇷훐탄떽룷떽듯뿍뮧1뫳,킶쿂뿍뮧1탨쟳뗄믵컯,좻뫳ힰ짏뿍뮧뗄뻠샫캪d0(뿍뮧1릩펦뗄믵컯,헢쪱튪싺ퟣu1+q3+qjj=1,2,ꆭ,L),퓙짨nk캪뗚k6ꇜQ;첨뎵솾엤쯍뗄뿍뮧쫽(nk=0뇭쪾캴쪹폃뗚k첨ꋛ뎵솾떽듯뿍뮧3쪱,킶쿂뿍뮧3탨쟳뗄믵컯,좻뫳뎵솾),벯뫏Rk뇭쪾뗚k쳵슷뺶,웤훐뗄뗚i룶풪ힰ짏뿍뮧3릩펦뗄믵컯,헢쪱튪싺ퟣu1+u3+q6ꇜ쯘rkQ;ꋜ뎵솾떽듯뿍뮧6쪱,킶쿂뿍뮧6탨튪뗄믵컯,ힰi뇭쪾뿍뮧rki퓚슷뺶k훐뗄쮳탲캪i(늻냼삨컯쇷훐탄),rk0=0뇭쪾컯쇷훐탄,좡쒿뇪몯쫽캪엤짏뿍뮧6릩펦뗄믵컯,헢쪱튪싺ퟣu1+u3+u6ꇜQ.쯍ퟜ샯돌ퟮ뛌,퓲뿉붨솢죧쿂ힰ킶믬뫏뎵솾슷뺶쪽(5)놣횤쎿쳵엤쯍슷뺶뗄뎤뛈늻뎬맽뎵솾튻듎컊쳢뗄믹폚횱맛쏨쫶뗄쫽톧쒣탍엤쯍뗄ퟮ듳탐쪻뻠샫.쪽(6)뇭쏷쎿쳵슷뺶짏뗄n뿍뮧쫽늻뎬맽ퟜ뿍뮧쫽.쪽(7)뇭쏷쎿룶뿍뮧뚼ꆡMinZ=ꇆKdr+dkr(n(i-1r)kiknrsignkk0k)](1)뗃떽엤쯍럾컱.쪽(8)뇭쪾쎿쳵슷뺶뗄뿍뮧뗄ퟩk=1[ꇆki=1n돉.쪽(9)쿞훆쎿룶뿍뮧평튻첨뎵솾럾컱.쪽(10)ꆡ.ꇆkqrꇜQkk(2)i뇭쪾떱뗚k솾뎵럾컱뗄뿍뮧쫽ꇝ1쪱,쮵쏷룃첨i=1nꇆj뎵닎볓쇋엤쯍,퓲좡sign(nk)=1,떱뗚k솾뎵럾ur+kiꇆkqrꇜQkki컱뗄뿍뮧쫽<1쪱,뇭쪾캴쪹폃룃첨뎵솾,틲듋좡i=1i=j+1ꆡꆡ(j=1,2,ꆭ,nk-1)(3)sign(nk)=ꇆk짏쫶ힰ킶믬뫏뎵솾슷뺶컊쳢뗄믹폚횱맛쏨쫶urꇜQkk(4)i뗄쫽톧쒣탍폫쿠맘퇐뺿컄쿗훐믹폚췸싧춼뗄쒣탍i=1nꇆk쿠뇈,뻟폐틔쿂쳘뗣:1)쒿뇪몯쫽뫍풼쫸쳵볾뷏dr+dgn(nk)ꇜDk(5)k(i-1r)kriknrsikk0캪좫쏦뫍뷓뷼쪵볊;2)뻶닟뇤솿ꆢ쒿뇪몯쫽뫍풼i=10ꇜnkꇜL(6)쫸쳵볾뗄뇭쪾뷏캪ퟔ좻ꆢ횱맛뫍틗폚샭뷢;3)뇣ꇆK폚짨볆쟳뷢쯣램뫍폃볆쯣믺뇠돌쟳뷢;4)뻟폐뷏nk=L(7)뫃뗄뿉삩돤탔.k=1Rk={rki|rkiꇊ{1,2,ꆭ,L},2ꆡힰ킶믬뫏뎵솾슷뺶컊쳢뗄쒣쓢췋믰ꆡꆡi=1,2,ꆭ,nk}(8)쯣램RkꇉR=ªꆡΠk1k21ꇙk2(9)1ꆡnkꇝꆡ쒣쓢췋믰쯣램뗄풭샭뫍쪵쿖늽훨sign(nk)=0ꆡ웤쯻(10)췋믰쫇튻훖컯샭맽돌,튻훖뷰쫴컯쳥볓죈훁짏쫶쒣탍훐,쪽(1)ꆢ(2)ꆢ(6)ꆢ(7)ꆢ(8)ꆢ(9)ꆢ튻뚨뗄컂뛈뫳,쯼뗄쯹폐럖ퟓ퓚웤ힴ첬뿕볤훐ퟔ(10)뷨본쇋컄쿗[16]붨솢뗄떥쿲뎵솾슷뺶컊쳢평퓋뚯.쯦ퟅ컂뛈뗄쿂붵,헢킩럖ퟓ훰붥춣쇴퓚늻뗄쫽톧쒣탍,짔폐룄뚯.쪽(3)ꆢ(4)ꆢ(5)쫇ퟷ헟룹춬뗄ힴ첬.퓚컂뛈ퟮ뗍쪱,럖ퟓ훘탂틔튻뚨뗄뷡릹뻝ힰ킶믬뫏뎵솾슷뺶컊쳢뗄풼쫸쳵볾붨솢뗄.쪽업쇐[17].춳볆솦톧뗄퇐뺿뇭쏷,퓚춬튻룶컂뛈,럖(1)캪쒿뇪몯쫽.쪽(2)ꆢ(3)ꆢ(4)폃폚놣횤뎵솾퓚ퟓ춣쇴퓚쓜솿킡뗄ힴ첬뗄룅싊뇈춣쇴퓚쓜솿듳뗄엤쯍맽돌훐웤퓘믵솿쪼훕늻뎬맽웤퓘훘솿,쿖뻙ힴ첬뗄룅싊튪듳.떱컂뛈쿠떱룟쪱,쎿룶ힴ첬뗄룅샽쮵쏷횮:짨쒳쳵엤쯍슷쿟짏폐죽룶뿍뮧,웤럾컱싊믹놾쿠춬,뚼뷓뷼욽뻹횵.떱컂뛈쟷쿲0쪱,럖쮳탲캪1ꆢ3ꆢ6,평튻첨퓘훘솿캪Q뗄엤쯍뎵솾췪ퟓ춣쇴퓚ퟮ뗍쓜솿ힴ첬뗄룅싊쟷쿲폚1.돉쯍믵뫍좡믵죎컱,본폚뎵솾퓚엤쯍맽돌훐뗄퓘쒣쓢췋믰쯣램쫇튻훖믹폚짏쫶췋믰풭샭붨솢믵솿쫇쯦ퟅ쯍믵뫍좡믵죎컱뗄췪돉뛸늻뛏뇤뮯뗄쯦믺쯑쯷쯣램.ퟩ뫏폅뮯컊쳢폫뷰쫴컯쳥뗄췋
ꆪ488ꆪ쾵ꆡ춳ꆡ릤ꆡ돌ꆡ톧ꆡ놨ꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡ뗚20뻭믰맽돌뿉뷸탐죧쿂샠뇈:ퟩ뫏폅뮯컊쳢뗄뷢샠쯆뷶뷏캪뢴퓓,뛸쟒늻틗샭뷢;2)뷢훐뗄풪쯘뷏뛠,폚뷰쫴컯쳥뗄ힴ첬,ퟩ뫏폅뮯컊쳢뗄ퟮ폅뷢샠쯆늻뷶햼폃뗄볆쯣믺듦뒢솿뷏듳,뛸쟒쟳뷢쪱볆쯣폚뷰쫴컯쳥쓜솿ퟮ뗍뗄ힴ첬,ퟩ뫏폅뮯컊쳢뗄럑쪱볤뷏뎤,쯑쯷킧싊뷏뗍;3)쯣램뗼듺맽돌훐닺폃몯쫽샠쯆폚뷰쫴컯쳥뗄쓜솿.짺뗄뷢훐늻뿉탐뷢뫜뛠,듳듳펰쿬쇋쯣램뗄쫕솲캪쇋뿋럾뻖늿쯑쯷쯣램벫틗쿝죫뻖늿ퟮ폅뷢쯙뛈.뗄좱뗣,쒣쓢췋믰쯣램쪹폃믹폚룅싊뗄쮫랽쿲쯦헫뛔짏쫶좱뗣,룹뻝ힰ킶믬뫏뎵솾슷뺶컊쳢믺쯑쯷벼쫵:떱믹폚쇚폲뗄튻듎닙ퟷ쪹떱잰뷢뗄뗄쳘뗣,놾컄쳡돶쇋튻훖탂뗄뷢뗄뇭쪾랽램,벴뿍훊솿쳡룟쪱,쒣쓢췋믰쯣램뷓쫜헢룶놻룄뷸뗄뷢뮧횱뷓업쇐뗄뷢뗄뇭쪾랽램.뛔폚튻룶폐L룶뿍ퟷ캪탂뗄떱잰뷢;퓚쿠랴뗄쟩뿶쿂,쯣램틔튻뚨뗄뮧뗄ힰ킶믬뫏뎵솾슷뺶컊쳢,헢훖뷢뗄뇭쪾랽램룅싊exp(-ꚤc/T)뷓쫜쿠뛔폚떱잰뷢살쮵훊솿뷏쫇횱뷓닺짺L룶1ꆫL볤뗄뮥늻훘뢴뗄ퟔ좻쫽뗄닮뗄뷢ퟷ캪탂뗄떱잰뷢,웤훐ꚤc캪쇚폲닙ퟷ잰뫳업쇐,뇭쪾뿍뮧쮳탲,좻뫳내헕ힰ킶믬뫏뎵솾슷뺶뷢뗄훊솿닮,T캪췋믰맽돌뗄뿘훆닎쫽(벴컂뛈).컊쳢뗄풼쫸쳵볾(냼삨뎵솾뗄퓘훘솿풼쫸뫍뎵솾쒣쓢췋믰쯣램틑퓚샭싛짏놻횤쏷쫇튻훖틔룅싊1튻듎엤쯍뗄ퟮ듳탐쪻뻠샫풼쫸),틀듎붫뷢뗄풪쯘쫕솲폚좫뻖ퟮ폅뷢뗄좫뻖폅뮯쯣램.(뿍뮧)뮮죫룷쳵엤쯍슷뺶훐.닉폃헢훖뇭쪾랽램쒣쓢췋믰쯣램뗄쪵쿖늽훨(틔쒿뇪몯쫽쟳ퟮ쪱,죴쒳룶뷢뛔펦뗄엤쯍슷뺶쫽듳폚엤쯍뎵솾ퟜ킡캪샽):첨쫽,퓲쮵쏷룃뷢캪튻룶늻뿉탐뷢.뗚1늽ꆡ톡뚨튻룶돵쪼뷢x0,쇮떱잰뷢xi=믹폚짏쫶뿍뮧횱뷓업쇐뗄뷢뗄뇭쪾랽램,뷡x0,떱잰뗼듺늽쫽k=0,떱잰컂뛈tk=tmax.뫏ힰ킶믬뫏뎵솾슷뺶컊쳢뗄쳘뗣,ퟷ헟릹퓬쇋쟳뗚2늽ꆡ죴퓚룃컂뛈듯떽쓚톭뮷춣횹쳵볾,뷢룃컊쳢뗄튻훖탂뗄쒣쓢췋믰쯣램,웤쯣램닟싔퓲뗚3늽,럱퓲,듓쇚폲N(x)i훐쯦믺톡튻쇚뻓죧쿂:xj,볆쯣ꚤfij=f(x)j-f(x)i,죴ꚤfijꇜ0,퓲xi=1)뷢뗄움볛랽램.폃쒣쓢췋믰쯣램쟳뷢ퟩ뫏x폅뮯컊쳢쪱,탨튪뛔뷢뷸탐움볛,쪹쯣램퓚뗼듺맽j,럱퓲,죴exp(-ꚤfij/tk)>random(0,1)(뇭쪾튻룶0떽1횮볤뗄뻹퓈쯦믺쫽),퓲xi=xj,훘뢴뗚돌훐,늻뛏쯑쯷떽훊솿룼폅뗄뷢.룹뻝뛔ힰ킶믬뫏2늽.뎵솾슷뺶컊쳢뗄쏨쫶,뛔폚쒳룶뷢튪에뚨웤폅쇓,뗚3늽ꆡkꇻk+1,tk+1=d(tk)(뇭쪾컂뛈쫗쿈튪뗃떽룃뷢쯹뛔펦뗄엤쯍슷뺶랽낸,좻뫳튪쿂붵뗄몯쫽),죴싺ퟣ훕횹쳵볾,뗚4늽,럱퓲,에뛏룃엤쯍슷뺶랽낸쫇럱싺ퟣ컊쳢뗄풼쫸쳵볾,뗚2늽.춬쪱볆쯣룃엤쯍슷뺶랽낸뗄쒿뇪몯쫽횵,퓚싺ퟣ뗚4늽ꆡ쫤돶볆쯣뷡맻,춣횹.컊쳢뗄풼쫸쳵볾뗄잰쳡쿂,웤쒿뇪몯쫽횵풽폅,뷢퓚짏쫶쒣쓢췋믰쯣램훐,냼몬튻룶쓚톭뮷뫍뗄훊솿풽룟.닉폃뿍뮧횱뷓업쇐뗄뇭쪾랽램닺짺튻룶췢톭뮷.쓚톭뮷캪뗚2늽,쯼뇭쪾퓚춬튻룶컂뗄뷢쯹좷뚨뗄엤쯍슷뺶랽낸,틾몬쓜릻싺ퟣ쎿룶뛈tk쪱퓚튻킩ힴ첬쯦믺쯑쯷.췢톭뮷훷튪냼삨뗚뿍뮧뚼뗃떽엤쯍럾컱벰쎿룶뿍뮧뷶평튻첨뎵솾럾3늽뗄컂뛈쿂붵뇤뮯tk+1=d(tk),뗼듺늽쫽뗄퓶컱뗄풼쫸,튲쓜싺ퟣ뎵솾뗄퓘훘솿풼쫸벰뎵솾튻볓kꇻk+1뫍춣횹쳵볾.듎엤쯍뗄ퟮ듳탐쪻뻠샫뗄풼쫸,떫늻쓜놣횤엤쯍ꆡힰ킶믬뫏뎵솾슷뺶컊쳢뗄쒣쓢췋믰쯣램뗄슷뺶쳵쫽킡폚엤쯍뎵솾ퟜ첨쫽.뛔폚쒳룶뷢,짨웤짨볆뛔펦뗄엤쯍슷뺶랽낸뗄엤쯍슷뺶쳵쫽폫엤쯍뎵솾폃쒣쓢췋믰쯣램쟳뷢ힰ킶믬뫏뎵솾슷뺶컊쳢ퟜ첨쫽횮닮캪M(죴엤쯍슷뺶쳵쫽ꇜ엤쯍뎵솾ퟜ쪱,좷뚨뷢뗄뇭쪾랽램쫇튻쿮럇뎣맘볼뗄릤ퟷ,쯼횱첨쫽,퓲좡M=0,뇭쪾룃뷢캪뿉탐뷢;죴엤쯍슷뷓뻶뚨쯣램쪵쿖뗄쓑틗돌뛈뫍쯣램탔쓜뗄폅쇓.뺶쳵쫽>뎵솾ퟜ첨쫽,퓲M>0,뇭쪾룃뷢캪튻쿖폐컄쿗폃쒣쓢췋믰쯣램쟳뷢뎵솾슷뺶컊쳢룶늻뿉탐뷢),웤쒿뇪몯쫽횵캪Z,뿉붫M뾴돉룃쪱뛠닉폃폐쿲뇟업쇐뗄뷢뗄뇭쪾랽램,믹폚헢훖뷢뛔펦뗄엤쯍슷뺶랽낸뗄늻뿉탐슷뺶쳵쫽,짨뛔뷢뗄뇭쪾랽램쯹릹퓬뗄쒣쓢췋믰쯣램뻟폐틔쿂좱쎿쳵늻뿉탐슷뺶뗄돍랣좨훘캪Pw(룃좨훘뿉룹뻝뗣:1)평폚뷢뗄뇭쪾늻첫횱맛,쪹뗃쯣램닟싔늻쒿뇪몯쫽뗄좡횵랶캧좡튻룶쿠뛔뷏듳뗄헽쫽),퓲
뗚5웚ꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡ색쎯쿩:ힰ킶믬뫏뎵솾슷뺶컊쳢뗄쒣쓢췋믰쯣램퇐뺿ꆪ489ꆪ룃뷢뗄움볛횵E뿉폃릫쪽(11)볆쯣.탐뷢,틲뛸쯣램뗄쫕솲쯙뛈뷏뿬.E=Z+MPw(11)2)쇚폲닙ퟷ랽램.쒣쓢췋믰쯣램쫇튻훖믹폚3ꆡ쪵퇩볆쯣뫍뷡맻럖컶쇚폲쯑쯷벼쫵뗄쯣램,좷뚨쇚폲닙ퟷ랽램쫇릹퓬룃쯣램뗄튻룶훘튪늽훨.놾컄닉폃솽붻뮻램쪵쪩쇚폲닙ퟷ.룃랽램쫇횸쯦믺톡퓱뷢훐뗄솽룶풪쯘샻폃쒣쓢췋믰쯣램뛔튻룶평볆쯣믺쯦믺닺짺,늢붻뮻웤횵뗄쇚폲닙ퟷ랽램.쿖뻙샽쮵쏷횮:짨쒳뗄ힰ킶믬뫏뎵솾슷뺶컊쳢쪵샽뷸탐쇋쪵퇩볆쯣.뷢캪샽1ꆡ짨컯쇷훐탄뫍20룶뿍뮧럖늼퓚튻룶뇟S=123456789,쯦믺닺짺뗄붻뮻뗣캪뗚4캻뫍뗚7캻,퓲쪵쪩솽붻뮻뫳뿉뗃풭뷢뗄튻룶쇚뻓뎤캪20km뗄헽랽탎뗘폲쓚,쎿룶뿍뮧뗄믵컯탨쟳솿뫍릩펦솿뻹퓚2t틔쓚(몬),컯쇷훐탄폐10첨Sꆯ=)붵컂랽램뗄좷뚨.컂뛈맜샭쫇쒣쓢췋믰쯣엤쯍뎵솾,뎵솾뗄퓘훘솿뻹캪8t,뎵솾튻듎엤쯍램쓑틔뒦샭뗄컊쳢횮튻,닉폃tk+1=atk탎쪽뗄뗄ퟮ듳탐쪻뻠샫뻹캪50km.샻폃볆쯣믺쯦믺닺붵컂랽램(a캪붵컂쾵쫽),벴쎿튻늽컂뛈틔쿠춬짺쇋컯쇷훐탄뫍20룶뿍뮧뗄캻훃ퟸ뇪틔벰룷뿍뗄뇈싊쿂붵뮧뗄믵컯탨쟳솿뫍릩펦솿,웤훐컯쇷훐탄뗄ퟸ뇪.4)쎿튻컂뛈뗄뗼듺늽뎤뗄좷뚨.닉폃만뚨뗄캪(,),20룶뿍뮧뗄ퟸ뇪벰웤믵컯뗼듺늽뎤,벴퓚쎿튻컂뛈뗼듺쿠춬뗄늽쫽,룃늽쫽탨쟳솿뫍릩펦솿볻뇭1.튪쟳룹뻝짏쫶쳵볾,뫏샭펦룹뻝컊쳢뗄맦쒣좷뚨.낲업뎵솾뗄엤쯍슷쿟,쪹엤쯍ퟜ샯돌ퟮ뛌.캪볲뇣5)훕횹ힼ퓲뗄좷뚨.닉폃컂뛈쿂붵횸뚨듎쫽웰볻,짨룷뿍뮧쿠뮥횮볤벰컯쇷훐탄폫뿍뮧횮볤뫳춣횹볆쯣뗄훕횹ힼ퓲.캪횱쿟뻠샫,룃뻠샫뿉룹뻝뿍뮧뫍컯쇷훐탄뗄ퟸ짏쫶믹폚뿍뮧횱뷓업쇐뗄뷢뗄뇭쪾랽램뗄쒣뇪볆쯣뗃떽.쓢췋믰쯣램뻟폐틔쿂폅뗣:1)뷢뗄뇭쪾쪮럖횱룹뻝샽1뗄쳘뗣,퓚폃쒣쓢췋믰쯣램뛔웤쟳맛,쟒쯣램닟싔뷏캪볲떥뫍틗폚샭뷢;2)뷢훐뗄뷢쪱닉폃쇋틔쿂퓋탐닎쫽:뛔늻뿉탐슷뺶뗄돍랣풪쯘짙,햼폃뗄볆쯣믺듦뒢솿킡,쟒볆쯣킧싊뷏좨훘좡300km,돵쪼컂뛈좡500,ퟜ붵컂듎쫽좡룟;3)평폚헢훖뷢뗄뇭쪾랽램돤럖샻폃쇋컊쳢뗄400,쎿듎붵컂뗄뗼듺늽뎤좡40,붵컂쾵쫽좡.횪쪶,쪹뗃쯣램퓚뗼듺맽돌훐닺짺뗄뷢듳뛠캪뿉뛔샽1쯦믺쟳뷢10듎,뗃떽뗄볆쯣뷡맻볻뇭2.뇭1ꆡ샽1뗄틑횪쳵볾뿍뮧뇠뫅12345678910뫡ퟸ뇪x/ퟝퟸ뇪/믵컯탨쟳솿q/믵컯릩펦솿u/뿍뮧뇠뫅11121314151617181920뫡ퟸ뇪x/ퟝퟸ뇪/믵컯탨쟳솿q/믵컯릩펦솿u/뇭2ꆡ닉폃쮫쿲엤쯍닟싔쪱샽1뗄쒣쓢췋믰쯣램볆쯣뷡맻볆쯣듎탲12345678910욽뻹엤쯍ퟜ뻠샫/쪹폃뗄뎵솾쫽44444444444쫗듎쯑쯷떽ퟮ훕뷢뗄붵컂듎쫽볆쯣쪱볤/
ꆪ490ꆪ쾵ꆡ춳ꆡ릤ꆡ돌ꆡ톧ꆡ놨ꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡ뗚20뻭ꆡꆡ듓뇭2뿉틔뾴돶:폃쒣쓢췋믰쯣램뛔샽1뗄뛔샽1튲뿉틔폃떥쿲엤쯍닟싔뷸탐쟳뷢,벴10듎쟳뷢훐,뚼뗃떽쇋훊솿뫜룟뗄뷢,웤엤쯍ퟜ냑샽1뾴돉솽룶떥쿲뎵솾슷뺶컊쳢,튻룶쫇붫뿍뻠샫뗄욽뻹횵캪,욽뻹쪹폃뗄뎵솾쫽캪뮧탨쟳뗄믵컯듓컯쇷훐탄쯍떽룷룶뿍뮧뗄뒿쯍믵4솾,뇈틑횪쳵볾욽뻹뷚쪡6솾뎵.웤훐뗚7듎뗃컊쳢,볲돆쯍믵컊쳢;쇭튻룶쫇붫뿍뮧릩펦뗄믵컯떽뗄뷢뗄훊솿ퟮ룟,웤엤쯍ퟜ샯돌캪,룃듓뿍뮧좡떽컯쇷훐탄뗄뒿좡믵컊쳢,볲돆좡믵컊ퟮ뫃뷢뛔펦뗄4쳵엤쯍슷뺶럖뇰캪:슷뺶1:0ꆪ쳢.ퟷ헟샻폃떥쿲뎵솾슷뺶컊쳢뗄쒣쓢췋믰쯣램6ꆪ15ꆪ2ꆪ16ꆪ3ꆪ11ꆪ10ꆪ18ꆪ0;슷뺶2:0ꆪ뛔샽1뗄쯍믵컊쳢뫍좡믵컊쳢럖뇰뷸탐쇋쟳20ꆪ1ꆪ19ꆪ17ꆪ4ꆪ5ꆪ13ꆪ0;슷뺶3:0ꆪ9ꆪ8ꆪ뷢[17],쟳뷢쪱돽돵쪼컂뛈닉폃250췢,웤쯼퓋탐14ꆪ7ꆪ0;슷뺶4:0ꆪ12ꆪ0.쒣쓢췋믰쯣램뗄볆쯣닎쫽죔닉폃폫짏쫶쮫쿲엤쯍닟싔쿠춬뗄닎쫽.뛔뷡맻튲뷏컈뚨,10듎뷢뗄뇪ힼ닮캪.듓볆쯍믵컊쳢뫍좡믵컊쳢럖뇰쯦믺쟳뷢10듎,뗃떽뗄쯣킧싊뾴,10듎쟳뷢뗄욽뻹볆쯣쪱볤뷶캪,욽뻹볆쯣뷡맻뫍ퟮ뫃볆쯣뷡맻볻뇭3.볆쯣킧싊뷏룟.붫뇭3훐쯍믵컊쳢뫍좡믵컊쳢뗄볆쯣뷡맻뫏평틔짏볆쯣뿉틔뾴돶:폃쒣쓢췋믰쯣램쟳뷢늢,벴뗃닉폃떥쿲엤쯍닟싔쪱샽1뗄쒣쓢췋믰쯣ힰ킶믬뫏뎵솾슷뺶컊쳢,뿉틔좡뗃뫜뫃뗄볆쯣뷡램볆쯣뷡맻.샽1뗄쮫쿲엤쯍볆쯣뷡맻폫떥쿲엤맻,쟒쯣램뗄볆쯣뷡맻뷏컈뚨,볆쯣킧싊뷏룟.쯍볆쯣뷡맻뗄뇈뷏볻뇭4.뇭3ꆡ샽1훐쯍믵컊쳢뫍좡믵컊쳢뗄쒣쓢췋믰쯣램볆쯣뷡맻컊쳢엤쯍ퟜ뻠샫뗄욽뻹쪹폃뗄욽뻹볆쯣ퟮ뫃뗄ퟮ뫃뗄뷢쪹폃뗄욽뻹횵/km뎵솾쫽쪱볤/s뷢/km뎵솾쫽쯍믵컊쳢좡믵컊쳢뇭4ꆡ샽1뗄쮫쿲엤쯍폫떥쿲엤쯍쒣쓢췋믰쯣램볆쯣뷡맻뗄뇈뷏엤쯍닟싔엤쯍ퟜ뻠샫뗄욽뻹쪹폃뗄욽뻹볆쯣ퟮ뫃뗄뷢ퟮ뫃뗄뷢쪹폃뗄욽뻹횵/km뎵솾쫽쪱볤/s/km뎵솾쫽쮫쿲엤쯍떥쿲엤쯍쮫쿲엤쯍닟싔뇈떥쿲엤쯍닟싔뷚쪡
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