CandidateBenediktBienhulsUniPiESRDays2020November232020120Outline1MathematicalModels2AdaptiveLargeNeighborhoodSearch3ConstraintShortestPathProblem4ALNSextensionsCandidateBenediktBienhulsUniPiESRDays2 ID: 878624
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1 IntegratingPlanningandReal-TimeOperation
IntegratingPlanningandReal-TimeOperationsinPublicTransportSystemsUNIVERSITADIPISADIPARTIMENTODIINFORMATICASUPERVISORMENTORProf.AntonioFrangioniSamuelaCarosiUNIVERSITADIPISAM.A.I.O.R. Candidate:BenediktBienh¨uls(UniPi) ESRDays2020 November23,20201/20 Outline 1 MathematicalModels 2 AdaptiveLargeNeighborhoodSearch 3 ConstraintShortestPathProblem 4 ALNSextensions Candidate:BenediktBienh¨uls(UniPi) ESRDays2020 November23,20202/20 Outline 1 MathematicalModels 2 AdaptiveLargeNeighborhoodSearch 3 ConstraintShortestPathProblem 4 ALNSextensions Candidate:BenediktBienh¨uls(UniPi) ESRDays2020 November23,20203/20 Timetablingbasedonthesetofallplannedtrips: qt2f0;1gt2VTTr;r
2 2R; 1 Timetablingbasedonthesetofallpoten
2R; 1 Timetablingbasedonthesetofallpotentialtrips: P(j;i)2ATTrzjiP(i;j)2ATTrzij=8]TJ ; -1; .63; Td; [00;:1;ifi=vr;1;ifi=v+r;0;else,i2VTTr;r2R:zij2f0;1g(i;j)2ATTr;r2R; 2 LinkingconstraintsTTVSDS: Pd2DPe2EpairdP(i;j)2AVSDSe(t)xeij=1qtt2^TRTWr;r2R; 1 Pd2DPe2EpairdP(i;j)2AVSDSe(t)xeij=X(t;k)2ATTrztkt2Tr;r2R; 2 Vehicle-DriverpairSchedulingCompact(withoptionaltime-windowarcconstraints): P(j;i)2AVSDSexejiP(i;j)2AVSDSexeij=8]TJ ; -1; .63; Td; [00;:1;ifi=s;1;ifi=s+;0;else,i2VVSDSe;e2Epaird;d2D;Pd2DPe2EpairdP(i;j)2AVSDSd(q)xeijwqq2Q;Wse1P(i;j)2AVSDSewsijxeijWse2s2Se;e2Epaird;d2D;xeij2f0;1g(
3 i;j)2AVSDSe;e2Epaird;d2D: 1 xekixei
i;j)2AVSDSe;e2Epaird;d2D: 1 xekixeij(k;i)2ATWe;(i;j)2AVSDSe(t);t2^TRTWe;e2Epaird;d2D;xekixejl(k;i)2ATWe;(i;j)2AVSDSe(t);(j;l)2APBe;t2^TRTWe;e2Epaird;d2D; a Pd2DPe2EpairdPu2Ue(t)xeu=1qtt2^TRTWr;r2R; 3 Pd2DPe2EpairdPu2Ue(t)xeu=X(t;k)2ATTrztkt2VTTr;r2R; 4 Vehicle-DriverpairSchedulingSetPartitioning: Pu2Ud(e)xdu=1;e2Epaird;d2D;Pd2DPu2Ud(q)xduwqq2Q:xdu2f0;1gu2Ud;d2D; 2 Candidate:BenediktBienh¨uls(UniPi) ESRDays2020 November23,20204/20 Timetablingbasedonthesetofallplannedtrips: qt2f0;1gt2VTTr;r2R; 1 Timetablingbasedonthesetofallpotentialtrips: P(j;i)2ATTrzjiP(i;j)2ATTrzij=8]TJ ; -1; .63; Td; [00;:1;ifi=vr;1;ifi=v+r;0;else
4 ,i2VTTr;r2R:zij2f0;1g(i;j)2ATTr;r2R; 2 L
,i2VTTr;r2R:zij2f0;1g(i;j)2ATTr;r2R; 2 LinkingconstraintsTTVS: Pd2D(t)P(i;j)2AVSd(t)xdij=1qtt2^TRTWr;r2R; 1 Pd2DPe2Epaird(t)xeij=P(t;k)2ATTrztkt2Tr;r2R; 2 VehicleSchedulingCompact(withoptionaltime-windowarcconstraints): P(j;i)2AVSdxdjiP(i;j)2AVSdxdij=8]TJ ; -1; .63; Td; [00;:S;ifi=s;S+;ifi=s+;0;else,i2VVSd;d2D;P(j;i)2AVSfxfjiP(i;j)2AVSfxfij=8]TJ ; -1; .63; Td; [00;]TJ ; -1; .63; Td; [00;:1;ifi=s;1;ifi=s+;0;else,i2VVSf;f2Frefd;d2D;Pd2DP(i;j)2AVSd(q)xdij+Pd2DPf2FrefdP(i;j)2AVSf(q)xfijwqq2Q:xdijwdij;xdij2N0(i;j)2AVSd;d2D:xfij2f0;1g(i;j)2AVSf;f2Frefd;d2D: 1 xdkixdij(
5 k;i)2ATWd;(i;j)2AVSd(t);t2^TRTWd;d2D;xdk
k;i)2ATWd;(i;j)2AVSd(t);t2^TRTWd;d2D;xdkixdjl(k;i)2ATWd;(i;j)2AVSd(t);(j;l)2APBd;t2^TRTWd;d2D; a LinkingconstraintsVSDS: Pe2Ed(i;j)yeij=xdij(i;j)2AVSDSd;d2D; 1 Pe2Ed(i;j)yeij=Pu2Ud(i;j)xdu(i;j)2AVSDSd;d2D; 2 DriverSchedulingCompactwithoptionaltime-windowarcconstraints: P(j;i)2ADSeyejiP(i;j)2ADSeyeij=0i2VDSe;e2Ed;d2D;Wse1P(i;j)2ADSewsijyeijWse2s2Se;e2Ed;d2D;yeij2f0;1g(i;j)2ADSe;e2Ed;d2D; 1 yekiyeij(k;i)2ATWe;(i;j)2ADSe(t);t2^TRTWe;e2Ed;d2D;yekiyejl(k;i)2ATWe;(i;j)2ADSe(t);(j;l)2APBe;t2^TRTWe;e2Ed;d2D; a Pd2DPe2EdP(i;j)2ADSeyeij=P(t;k)2ATTrztki2VDS(t);t2VTTr;r2R; 1 Pd2DPu2Ud(t)xdu=1qtt2^TRTWr;r2R; 3 Pd2DPu2Ud(t)xdu=P(t;k)2ATTrztkt2VTTr;r2R;
6 4 VehicleSchedulingSetPartitioning: Pu2U
4 VehicleSchedulingSetPartitioning: Pu2Ud(s)xdu=S;(s;S)2Sd;d2D;Pu2Ud(s+)xdu=S+;(s+;S+)2S+d;d2D;Pu2Ud(f)xdu=1;f2Frefd;d2D;Pd2DPu2Ud(q)xduwqq2Q:xdu2f0;1gu2Ud;d2D; 2 Pe2EdPu2Ue(i;j)yeu=Pu2Ud(i;j)xdu(i;j)2AVSDSd;d2D; 4 Pe2EdPu2Ue(i;j)yeu=xdij(i;j)2AVSDSd;d2D; 3 DriverSchedulingSetPartitioning: Pu2Ud(e)ydu=1;e2Ed;d2D;yeu2f0;1gu2Ue;e2Ed;d2D: 2 LinkingconstraintsTSDS(graphconsideringconstrainedvehiclechanges): Pd2DPe2EdP(i;j)2ADSePu2Ue(i;j)yeu=P(t;k)2ATTrztki2VDS(t);t2VTTr;r2R; 2 Candidate:BenediktBienh¨uls(UniPi) ESRDays2020 November23,20205/20 Outline 1 MathematicalModels 2 AdaptiveLargeNeighborhoodSearch 3 ConstraintShortestPathProblem 4 ALNSextensio
7 ns Candidate:BenediktBienh¨uls(UniPi) E
ns Candidate:BenediktBienh¨uls(UniPi) ESRDays2020 November23,20206/20 ALNS:DestroyOperatorsRandom:Arandomsetofvehicleanddriverpairsisselected.Lessfrequentchoices:Theleastfrequentlychosenpairsofthepreviousiterationsareselected.Linedirection:Selectpairsthatcontaintripsofarandomlychosendirectionareselected.Thisisdonetoenableshiftingofthecorrespondingtrips.Proximity:Thisdestroyoperatoraimsatincreasingthechanceofimprovementbyconsideringpairsthatareclosetoareferencepairerinbothtimeandspace.Wedenethisproximitybycalculatingmini2er;j2ewithe2Epairdnerfcij+(dtjati);cji+(dtiatj)g;withcijrepresentingthecostofconnectingtripiandjinthesameduty(cij=1iftheycannotbefe
8 asiblyconnected)anddtjatibeingthetim
asiblyconnected)anddtjatibeingthetimefromtheendoftripitothebeginningoftripj,whichisweighedbyafactor.Proximitytocancellation:Similartothepreviousoperator,butinsteadofareferencepairwecomparetheproximitytothecurrentlycancelledtrips. Candidate:BenediktBienh¨uls(UniPi) ESRDays2020 November23,20207/20 ALNS:RepairOperatorsRandom:Arandomsetofvehicleanddriverpairsisselected.Lessfrequentchoices:Theleastfrequentlychosenpairsofthepreviousiterationsareselected.Linedirection:Selectpairsthatcontaintripsofarandomlychosendirectionareselected.Thisisdonetoenableshiftingofthecorrespondingtrips.Proximity:Thisdestroyoperatoraimsatincreasingthechanceofimprovementbyconsideringpairsthat
9 areclosetoareferencepairerinbothtimeands
areclosetoareferencepairerinbothtimeandspace.Wedenethisproximitybycalculatingmini2er;j2ewithe2Epairdnerfcij+(dtjati);cji+(dtiatj)g;withcijrepresentingthecostofconnectingtripiandjinthesameduty(cij=1iftheycannotbefeasiblyconnected)anddtjatibeingthetimefromtheendoftripitothebeginningoftripj,whichisweighedbyafactor.Proximitytocancellation:Similartothepreviousoperator,butinsteadofareferencepairwecomparetheproximitytothecurrentlycancelledtrips. Candidate:BenediktBienh¨uls(UniPi) ESRDays2020 November23,20208/20 ALNS:RepairOperators Timetablingbasedonthesetofallplannedtrips: qt2f0;1gt2VTTr;r2R; 1 LinkingconstraintsTTVSDS: Pd2DPe2EpairdP(i;j)2AVSDSe
10 (t)xeij=1qtt2^TRTWr;r2R; 1 Vehicle-D
(t)xeij=1qtt2^TRTWr;r2R; 1 Vehicle-DriverpairSchedulingCompact(withoptionaltime-windowarcconstraints): P(j;i)2AVSDSexejiP(i;j)2AVSDSexeij=8]TJ ; -1; .63; Td; [00;:1;ifi=s;1;ifi=s+;0;else,i2VVSDSe;e2Epaird;d2D;Pd2DPe2EpairdP(i;j)2AVSDSd(q)xeijwqq2Q;Wse1P(i;j)2AVSDSewsijxeijWse2s2Se;e2Epaird;d2D;xeij2f0;1g(i;j)2AVSDSe;e2Epaird;d2D: 1 xekixeij(k;i)2ATWe;(i;j)2AVSDSe(t);t2^TRTWe;e2Epaird;d2D;xekixejl(k;i)2ATWe;(i;j)2AVSDSe(t);(j;l)2APBe;t2^TRTWe;e2Epaird;d2D; a Pd2DPe2EpairdPu2Ue(t)xeu=1qtt2^TRTWr;r2R; 3 Vehicle-DriverpairSchedulingSetPartitioning: Pu2Ud(e)xdu=1;e2Epaird;d2D;Pd2DPu2Ud(q)xduwqq2Q:xdu2f0
11 ;1gu2Ud;d2D; 2 Candidate:BenediktBienh¨
;1gu2Ud;d2D; 2 Candidate:BenediktBienh¨uls(UniPi) ESRDays2020 November23,20209/20 ALNS:RepairOperatorsRandom:Arandomsetofvehicleanddriverpairsisselected.Lessfrequentchoices:Theleastfrequentlychosenpairsofthepreviousiterationsareselected.Linedirection:Selectpairsthatcontaintripsofarandomlychosendirectionareselected.Thisisdonetoenableshiftingofthecorrespondingtrips.Proximity:Thisdestroyoperatoraimsatincreasingthechanceofimprovementbyconsideringpairsthatareclosetoareferencepairerinbothtimeandspace.Wedenethisproximitybycalculatingmini2er;j2ewithe2Epairdnerfcij+(dtjati);cji+(dtiatj)g;withcijrepresentingthecostofconnectingtripiandjinthesameduty(cij=1ifthe
12 ycannotbefeasiblyconnected)anddtjati
ycannotbefeasiblyconnected)anddtjatibeingthetimefromtheendoftripitothebeginningoftripj,whichisweighedbyafactor.Proximitytocancellation:Similartothepreviousoperator,butinsteadofareferencepairwecomparetheproximitytothecurrentlycancelledtrips. Candidate:BenediktBienh¨uls(UniPi) ESRDays2020 November23,202010/20 ALNS:RepairOperators Timetablingbasedonthesetofallpotentialtrips: P(j;i)2ATTrzjiP(i;j)2ATTrzij=8]TJ ; -1; .63; Td; [00;:1;ifi=vr;1;ifi=v+r;0;else,i2VTTr;r2R:zij2f0;1g(i;j)2ATTr;r2R; 2 LinkingconstraintsTTVSDS: Pd2DPe2EpairdP(i;j)2AVSDSe(t)xeij=X(t;k)2ATTrztkt2Tr;r2R; 2 Vehicle-DriverpairSchedulingCompact(withoptionaltime-w
13 indowarcconstraints): P(j;i)2AVSDSexeji&
indowarcconstraints): P(j;i)2AVSDSexejiP(i;j)2AVSDSexeij=8]TJ ; -1; .63; Td; [00;:1;ifi=s;1;ifi=s+;0;else,i2VVSDSe;e2Epaird;d2D;Pd2DPe2EpairdP(i;j)2AVSDSd(q)xeijwqq2Q;Wse1P(i;j)2AVSDSewsijxeijWse2s2Se;e2Epaird;d2D;xeij2f0;1g(i;j)2AVSDSe;e2Epaird;d2D: 1 xekixeij(k;i)2ATWe;(i;j)2AVSDSe(t);t2^TRTWe;e2Epaird;d2D;xekixejl(k;i)2ATWe;(i;j)2AVSDSe(t);(j;l)2APBe;t2^TRTWe;e2Epaird;d2D; a Pd2DPe2EpairdPu2Ue(t)xeu=X(t;k)2ATTrztkt2VTTr;r2R; 4 Vehicle-DriverpairSchedulingSetPartitioning: Pu2Ud(e)xdu=1;e2Epaird;d2D;Pd2DPu2Ud(q)xduwqq2Q:xdu2f0;1gu2Ud;d2D; 2 Candidate:BenediktBienh¨uls(UniPi) ESRDays2020 November23,202011
14 /20 Outline 1 MathematicalModels 2 Adapt
/20 Outline 1 MathematicalModels 2 AdaptiveLargeNeighborhoodSearch 3 ConstraintShortestPathProblem 4 ALNSextensions Candidate:BenediktBienh¨uls(UniPi) ESRDays2020 November23,202012/20 ConstraintShortestPathProblem minws0xs:t:P(j;i)2AxjiP(i;j)2Axij=8]TJ ; -1; .63; Td; [00;:1;ifi=s;1;ifi=s+;0;else,i2V;P(i;j)2AwsijxijWss2Sns0;xij2f0;1g(i;j)2A; max0minws0x+(X(i;j)2AwsijxijWs)s:t:P(j;i)2AxjiP(i;j)2Axij=8]TJ ; -1; .63; Td; [00;:1;ifi=s;1;ifi=s+;0;else,i2V;xij2f0;1g(i;j)2A; Candidate:BenediktBienh¨uls(UniPi) ESRDays2020 November23,202013/20 ConstraintShortestPathProblem minws0xs:t
15 :P(j;i)2AxjiP(i;j)2Axij=8
:P(j;i)2AxjiP(i;j)2Axij=8]TJ ; -1; .63; Td; [00;:1;ifi=s;1;ifi=s+;0;else,i2V;P(i;j)2AwsijxijWss2Sns0;xij2f0;1g(i;j)2A; max0minws0x+(X(i;j)2AwsijxijWs)s:t:P(j;i)2AxjiP(i;j)2Axij=8]TJ ; -1; .63; Td; [00;:1;ifi=s;1;ifi=s+;0;else,i2V;xij2f0;1g(i;j)2A; Candidate:BenediktBienh¨uls(UniPi) ESRDays2020 November23,202013/20 ConstraintShortestPathProblem MotivatedbyDumitrescuandBoland(2003) Preprocessingandallpairsshortestpathproblemstoreducecomputationalburden. Candidate:BenediktBienh¨uls(UniPi) ESRDays2020 November23,202014/20 ConstraintShortestPathProblem minws0xs:t:P(j;i)2Axji
16 P(i;j)2Axij=8]TJ ; -1;
P(i;j)2Axij=8]TJ ; -1; .63; Td; [00;:1;ifi=v;1;ifi=s+;0;else,i2V;P(i;j)2AwsijxijWss2Sns0;xij2f0;1g(i;j)2A; max0minws0x+(X(i;j)2Awsijxij(Ws))s:t:P(j;i)2AxjiP(i;j)2Axij=8]TJ ; -1; .63; Td; [00;:1;ifi=v;1;ifi=s+;0;else,i2V;xij2f0;1g(i;j)2A; Candidate:BenediktBienh¨uls(UniPi) ESRDays2020 November23,202015/20 ConstraintShortestPathProblem Algorithm1Preprocessing 1:Step0:2:Initialize:U0=(jVj1)maxi;j2Acij+1,U=U03:4:Step1:5:ComputeQc0jforallj2V6:ifnopathfrom0to1wasfoundthen7:ifU=U0then8:STOP:theproblemisinfeasible9:else10:STOP:thepathcorrespondingtoUisanoptimalsolution11:e
17 lse12:ifws(Qc0;1)Wsforalls2Sthen13:
lse12:ifws(Qc0;1)Wsforalls2Sthen13:ifc(Qc0;1)UthenQc0;1isanoptimalsolution14:elseSTOP:thepathcorrespondingtoUisanoptimalsolution15:else16:ifQc0;1Uthen17:STOP:thepathcorrespondingtoUisanopti-malsolution18:elseL=Qc0;119:Step2:20:fors2Sdo21:CalculateQLR(ws)0jforallj2V22:ifws(QLR(ws)0;1)Ws2then23:ifU=U0thenSTOP:theproblemisinfeasible24:elseSTOP:thepathcorrespondingtoUisanoptimalsolution25:elseifws0(QLR(ws)0;1)Ws08s02SthenU=c(QLR(ws)0;1)26:Step3:27:fori2Vnf0;1gdo28:ifQLR(ws)0;i+QLR(ws)i;1Wsforsomes2Sthen29:deletevertexiaswellasitsincidentarcs30:elseifQc0;i+Qci;1Uthen31:deletevertexiaswellasitsincidentarcs32:fori2f0;:::;jVj1gdo33:ifw
18 s(QLR(ws)0;i)+wsij+ws(QLR(ws)j;1)
s(QLR(ws)0;i)+wsij+ws(QLR(ws)j;1)Wsforsomes2Sthen34:delete(i;j)35:elseifc(Qc0;i)+cij+c(Qcj;1)Uthendelete(i;j)36:elseifws(Qc0;i)+wsij+ws(Qcj;1)Wsforalls2Sthen37:U=c(Qc0;i)+cij+c(Qcj;1)38:Step4:39:ifthegraphorUwaschangedinStep2or3thengotostep140:elseSTOP Candidate:BenediktBienh¨uls(UniPi) ESRDays2020 November23,202016/20 ConstraintShortestPathProblem Algorithm2LabelSettingAlgorithm Step0:RunAlgorithm1toobtainQci;1andQLR(ws)i;1foralls2Sandi2Vnf1gInitialize:L0=f(0;0)g;Li=;foralli2Vnf0gStep1:forj2Vnf1gintopologicalorderdofor(i;j)2Adoforlabels(w(Qli);c(Qli))2Lidoifws(Qli)+wsi;j+ws(QLR(ws)j;1)Wsforalls2Sandc(Qli)+cij+c(Qcj;1)UthenAddthecorrespondinglab
19 eltoLjwhilemaintain-ingalexicographicord
eltoLjwhilemaintain-ingalexicographicorderifws(Qli)+wsi;j+ws(Qcj;1)Wsforalls2SthenU=c(Qli)+cij+c(Qcj;1)RemovedominatedLabelsfromLj Candidate:BenediktBienh¨uls(UniPi) ESRDays2020 November23,202017/20 Outline 1 MathematicalModels 2 AdaptiveLargeNeighborhoodSearch 3 ConstraintShortestPathProblem 4 ALNSextensions Candidate:BenediktBienh¨uls(UniPi) ESRDays2020 November23,202018/20 ALNSextensions Retiming(Veelenturfetal.(2012),vanLieshoutetal.(2018)) FortripthatiscancelledafteraniterationofALNS,weintroduceshiftingopportunities.Ifallcancelledtripsalreadyhaveshiftingcopies,weallowshiftingfortripswithinasmallproximityofcancelledtrips. TripMerging(Gintneretal.(2005),Sevimeta
20 l.(2020)) Tripsthatareperformedinsuccess
l.(2020)) TripsthatareperformedinsuccessiononthesamevehicleanddriverpairinmultipleiterationsofALNSaremergedtoasingletrip.Thisonlyconsideredtripsofthedestroyedpartofthesolution. Candidate:BenediktBienh¨uls(UniPi) ESRDays2020 November23,202019/20 ReferencesI.DumitrescuandN.Boland.ImprovedPreprocessing,LabelingandScalingAlgorithmsfortheWeight-ConstrainedShortestPathProblem.Networks,42(3):135153,2003.ISSN00283045.doi:10.1002/net.10090.V.Gintner,N.Kliewer,andL.Suhl.Solvinglargemultiple-depotmultiple-vehicle-typebusschedulingproblemsinpractice.ORSpectrum,27(4):507523,2005.ISSN01716468.doi:10.1007/s00291-005-0207-9.s.Sevim,H.Tekiner-Mogulkoc¸,andM.G.G¨uler.Sc
21 hedulingthevehiclesofbusrapidtransitsyst
hedulingthevehiclesofbusrapidtransitsystems:acasestudy.InternationalTransactionsinOperationalResearch,00:125,2020.ISSN14753995.doi:10.1111/itor.12763.R.vanLieshout,J.Mulder,andD.Huisman.Thevehiclereschedulingproblemwithretiming.ComputersandOperationsResearch,96:131140,2018.ISSN03050548.doi:10.1016/j.cor.2018.04.008.URLhttps://doi.org/10.1016/j.cor.2018.04.008.L.P.Veelenturf,D.Potthoff,D.Huisman,andL.G.Kroon.Railwaycrewreschedulingwithretiming.TransportationResearchPartC:EmergingTechnologies,20(1):95110,2012.ISSN0968090X.doi:10.1016/j.trc.2010.09.008.URLhttp://dx.doi.org/10.1016/j.trc.2010.09.008. Candidate:BenediktBienh¨uls(UniPi) ESRDays2020 November23