Proceedings of the International MultiConference of Engineers and Comp - PDF document

Proceedings of the International MultiConference of Engineers and Computer Scientists 2012 Vol II, IMECS 2012, March 14 - 16, 2012, Hong Kong ISBN: 978-988-19251-9-0 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) IMECS 2012 SlackenPiercingPointBasedSubproblemAlgorithmfortheLPRelaxationofCrewRosteringZhaoMingyu,LiuQiongandWangZhenyu Algorithm1:ThemodiedalgorithmforCrewRosteringProblem 1A0 fajjt(aj)61;f(aj)2Cg[fajjf(aj)=2Cg; 02repeat 3Solvethesubproblemminz=cxs.t.A0x=b,letdenoteitsoptimaldualvector,andSCconsistsofcrewsthathavechangedinthelastbasis.4 max0;�dj dj�djdj0;dj&#x]TJ/;ཧ ;.96;d T; 9.;ܨ ;� Td;&#x [00;0;f(j)=2S5 
+(1�)
6Removeallnon-basiccolumnsfromA07ifz&#x]TJ/;ཧ ;.96;d T; 9.;ܨ ;� Td;&#x [00;bthen 8L ;9foreachaj=2A0^dj�0do 10ifdj=0then 11A0 A0[fajg12elseiff(aj)=2Sand(dj6+^dj�0)_(dj6�^dj0)then 13L L[fajg14endif15endforeach16AddthemostNscolumnstoA0whichpossesssmallreducecostwithinL17Addthemost(1�)NscolumnstoA0whichpossesssmallreducecostwithandtheninL18endif19until=0;20SPRINT() Especially,theemptyLoW(notincludeanytask)islegalforanycrewmember.ForanyLoWaj,itconsistsofanumberoftasks,anditscardinalityisdenotedast(aj).EachLoWajhasanassociatecostcjevaluatingitsfairness.Crewrosteringproblemistypicallyformulatedasanextensionofthesetpartitioningproblem,inwhichwewanttondaminimumcostsubsetoftheLoWsfollowingthreecategoriesofconstraints.TherstisthateachcrewmemberhastochooseoneandonlyoneLoWfromallLoWsbelong-ingtoher/him,whichisreferredtoasassignmentconstraint.Second,thecrewmembersassignedtoanytaskshouldnotexceeditscapacity.Third,sometasksmusthavecrewmembersmore/lessthansomeamount,whosequalicationshouldbeasdesire.EveryLoWcanformacolumnaccordingtotheseconstraints,andproperlyintroducessomesupplierandslackvariablesadditionally,withwhichtheconstraintmatrixAcanbeconstructed.AsalmosteverycolumninAindicatesaLoWaj,itisalsodenotedasajforsimplerepresentation.Decisionvariablexjisequalto1ifLoWajisincludedinthesolution,and0otherwise.Insummary,therelaxationofcrewrosteringproblemcanbeformulatedasaLP,see(1).Thereadersarereferredto[3]fordetailsonthereal-worldbusinessrulesandthecorrespondingrelationswiththemodel.mincx(P)s.t.Ax6bx�0(1)Withoutlossofgenerality,itcanbeassumedc�0andb�0.AsubsetofallLoWsandsomeothervariablesconstituteasubproblemof(P),whosesolutionisfeasibleto(P)anddenotedas(SP).III.THEMODIFIEDALGORITHMFORCRPInthissection,amodiedsubproblemalgorithmforCRPbasedonaslackenpiercingpointconceptwasproposed,withanefcientsubproblemconstructionstrategyadaptedtothisspecialproblem.TheformaldescriptionofthemodiedalgorithmwasgivenbyAlgorithm1.A.InitialsolutionIntherealworldenvironment,therosteringproblemhasn'tanyinitialfeasiblesolutionusually.TherstsubproblemconsistsofnotonlyallsingletaskLoWsandemptyLoWsforallcrew,butalsoallsupplierandslackvariables.Evidently,itcanproduceafeasiblesolutionfortheoriginalproblem,becauseitislegaltoassignemptyLoWtoallcrewmembersatleast.Inaddition,thevector0isadualfeasiblesolutionasthecostofanycolumnisnon-negative.B.SlackenpiercingpointAsmentionedbefore,themaindifferencebetweensub-problemalgorithmsishowtoconstructthenextsubproblemateachiteration.Likepriceschemeforthesimplex,differentsubproblemconstructionstrategieshavedramaticallyimpactontheoverallperformanceforthesubproblembasedmethod,asindicatedby[5],[6].Asaresult,thesubproblemconstruc-tionmethodiscertainlythekernelsubjectforasubproblembasedalgorithm.Letdenotetheincumbentdualvector,andtheoptimaldualsolutionofthecurrentsubproblem.Thepiercingpoint0canbewrittenas0=+(1�),wheremakes0remaindualfeasibleandimprovethedualobject0baslargeaspossible.Aprobeistheoperationofndingthepiercingpointsandthecolumnsthatdecide.Thosecolumnscouldberegardedasthebestcandidatesforthenextsubproblem.Theseconceptsweredenedby[7]withdualversion.Itiswellknownthatitispossibletodesignmoreefcientpricealgorithmsforaspecialclassofproblemswhichex-hibitsomeidentiablestructures.Fortherosteringproblem,everycrewmembermusthaveoneandonlyoneLoWinanyfeasiblesolution,asaconsequenceoftheassignmentconstraints.Thus,everycrewmemberhasseveralLoWsinanybasicfeasiblesolutionforitsLPrelaxation.Asforsomecrewmemberk2SC,ifher/hisLoWsincurrentbasishavechangedsincethelastiteration,weattempttokeeptheseLoWsinthenextbasis.Theideabehindisthatithas gotlocally“best”representationsforcrewk,soitisunlikelythatkisstillagoodimprovingcandidateinthenextiteration,i.e.thebestcandidatecolumnsareunlikelytocomefromthiscrewmember.So,thiscrewmemberwon'tbeconsideredintheprobeoperationanymore.Todistinguishwith,isdenedasbellow=max(0;�dj dj�djdj0;dj&#x]TJ/;ø 9;&#x.962; Tf;&#x 10.;Ԗ ;� Td;&#x [00;0;f(j)=2S)(2)wheredjanddjdenotereducecostsofcolumnajwithandrespectively.Theupdateddualvector0=+(1�)isreferredtoastheslackenpiercingpointofand.Ineachiteration,thecolumnsareregardedasthebestcandidateswhichhavezeroreducedcostwithslackenpiercingpoint.Thesecolumnslimitthefurtherimprovementofcurrentdualobject,sotheyarereferredtoasrestrictvariablesforsimplifyrepresentation.C.ColumnselectionBythedenitionofslackedpiercingpoint,itisevidentthatallrestrictvariablesshouldbeaddedtothesubproblem.Nevertheless,thereareonlyafewofrestrictvariablesineachiterationusually.Itnearlycan'tworkifonlytheserestrictvariablesareaddedtosubproblemforalargeproblem.Thus,wehavetochoosesomeotherpotentiallybenecialcolumns.Infact,asindicatedby[5],[6],thesizeofsubproblemhasanimportantimpactonsubproblembasedmethods,asthemorecolumnsareintroducedtothenextsubproblem,themoresolvingtimewillbeneededinevitably.Moreimportantly,atoogreedystrategybasedonlocalinformationwillleadtounnecessarysimplexiterationsfromaglobalview.Incontrast,toofewcolumnswillresultintoomanyredundantpricing.So,thesubproblemsizeshouldbetradedoffforthismodiedalgorithmcarefully.Althoughthegoaloftheslackenpiercingpointbasedcriteriaistosatisfythecomplementaryslacknessinasetwhichismadeofcolumnswithapproximatelyequalreducecostswith0,thecolumnswithsmallerreducedcostwith0arebetter.So,reducedcostwith0and0isusedtoevaluateothercolumnssimultaneously,ratherthantheP-Dsubproblemalgorithmdid.Inkeepingwithprobeoperation,othercolumnsofthecrewmembersthatbelongtoSwon'tbeaddedtosubproblemunlesstheyarerestrictvariables.AsforamembernotbelongingtoS,thecandidatesetLconsistsofcolumnswhosereducedcostwithislessthanzeroandreducedcostwithislessthanorequalto�,andthecolumnswhosereducedcostwithispositiveandreducedcostwithislessthanorequalto+,where+�.Thatis,L=najdj&#x-278;0;d0j6+andf(j)=2So[najdj0;d0j6�andf(j)=2So(3)Severalparametersareusedtocontrolthesubproblemsizeinthemodiedalgorithm.Toconstructthenextsubproblem,itisrstlyselectednomorethanNs(0661)columnswithsmallestreducedcostwith0fromL,regardlessofdj.Andthenitisaddedatmost(1�)Nscolumnswithsmallestreducedcostwith0anddj0fromL.Thatis,somecolumnswithdj0areusedinsteadofsomewithsmallerreducedcostwith0.ToefcientlyselectcolumnsfromL,twodimensionalbucketsareusedtorankcolumns,whichhaveapproximatelyequalreducecostswith0andplacedinthesamebucket.Ifah2Li;jandag2Li+1;jthendhdganddhdg,andifag2Li;j+1thendhdganddhdg.D.OptimalityNotethattheslackenpiercingpointdoesn'tconsiderallcolumnsintheprobeoperation,andthecorresponding0maynotbedualfeasibleforprimalproblem(P).Neverthe-less,ifthereducedcostofacolumnwiththeslackedpiercingpointisnegative,itisstillpossibletobecomenon-negativeinthelateriterations.Althoughitispossiblethatwecan'tobtaintheoptimumoftheoriginalproblembythismethodaftersatisfyingcondition=0,itisusualthattheincumbentisveryclosetoit.Ifacolumnnotdualfeasibleforexistsafterloop,calltheSPRINTalgorithmtoguaranteetheconvergence.AsfortheSPRINTalgorithm,itusesanarrayofappropriatesizetoaccommodatethepotentialcolumns.Thecolumnswithnegativereducedcostwithareappendedfromthebeginningofthearray,andotherslessthansomethreshold(suchas10�5)areappendedfromthebottomtoback.Thecolumnswithnegativereducedcostcanoverridetheothers,andnottrueinturn.Thesimplicationisrootedinthattheinitialbasisisveryclosetotheoptimalpoint.E.AnalysisThereareseveralreasonswhythisapproachiscompu-tationallyefcient.First,ituses6toupdate0intheprobeoperationsothatalargerdualobjectvalueisobtained,althoughitisslightlynotdualfeasiblefortheoriginalproblem(P).Atthesametime,itaddsmoregoodcolumnsforthecurrentsubproblem(dj0),whichwilldecreasetheobjectvalueofthenextsubproblemmorerapidly.Therefore,itconvergestosomegoodfeasiblepointfor(P)morequickly.Second,thisstrategylightenstheburdenontheprobeandsortoperationineachiteration,especiallybenecialtolargescaleproblem.Finally,evenacolumnmaybecomenotdualfeasibleinsomeiteration,itisnotpermanent.Itstillhasthepossibilityofbecomingdualfeasiblelater.TheconvergencepointoftheP-DphaseisusuallysoclosetotheoptimumoftheoriginalthatitcanrapidlyderiveoptimalitybySPRINTalgorithm.IV.COMPUTATIONALEXPERIENCEInthisstudy,anumberofproblemswereexperimentedona2:66GHzPCwith4GBofRAM,andXpress[8]wasusedastheLPsolverforthesubproblems.TheseproblemscamefromtheoperationcontrolsystemofalargeinternationalairlineinChina,andtargetedondifferentcrewranks.Infact,eachrankhasitsmodelcharacteristicsincethecorrespondingbusinessrulesareappropriative.TableIpresentsthenumbersofcrewsandtaskswiththescaleofconstraintmatrixintheproblemssolved.TableIIpresentseachoperation'stimeofthemodiedalgorithmforeveryproblemlistedinTableI.Everyproblem TABLEIIOPERATIONTIMEINTHEMODIFIEDALGORITHMFORTHEPROBLEMS Prob.MajorSimplexProbePriceSPRINTTotal iter.iter.timeupdatesortIter.iter.timeprice No.16615179615:082:733:544:23373921:230:2027:13No.212924500222:3510:4711:4911:1755:70No.37726029022:4511:1015:534:9737120:450:8655:63No.45439419938:9510:9415:877:2411266346:842:5082:80No.549986618:1713:8417:5811:0151:42No.65813860911:3323:0227:6513:752240:171:2077:83 TABLEIIIaCOMPARISONWITHP-DSUBPROBLEMSIMPLEXALGORITHMFORNO.13 No.1 No.2 No.3 Majoriterations 379 66 82:6% 429 129 69:9% 90 77 14:4%Solving(SP)time 22:56 15:08 33:2% 24:17 22:35 7:5% 33:23 22:45 32:4%Probetime 22:90 2:73 88:1% 43:13 10:47 75:7% 20:22 11:10 45:1%Updatetime 20:58 3:54 82:8% 38:15 11:49 69:9% 18:29 15:53 15:1%Sorttime 24:49 4:23 82:7% 36:99 11:17 68:8% 7:90 4:97 37:1%Probe+Sort/iter. 0:13 0:11 15:4% 0:19 0:17 10:5% 0:31 0:21 32:3%SPRINTphase 1:43 1:31 Total 90:77 27:13 70:1% 142:63 55:70 60:9% 79:88 55:63 30:4% TABLEIIIbCOMPARISONWITHP-DSUBPROBLEMSIMPLEXALGORITHMFORNO.46 No.4 No.5 No.6 Majoriterations 129 54 58:1% 107 49 54:2% 149 58 61:1%Solving(SP)time 57:10 38:95 31:8% 14:89 8:17 45:1% 19:43 11:33 41:7%Probetime 40:87 10:94 73:2% 41:47 13:84 66:6% 76:11 23:02 69:8%Updatetime 38:14 15:87 58:4% 38:91 17:58 54:8% 71:59 27:65 61:3%Sorttime 23:40 7:24 69:1% 36:03 11:01 69:4% 49:55 13:75 72:3%Probe+Sort/iter. 0:50 0:34 32:0% 0:72 0:51 29:2% 0:84 0:63 25:0%SPRINTphase 9:38 1:39 Total 159:95 82:80 48:2% 131:83 51:42 61:0% 217:36 77:83 64:2% TABLEIPROBLEMSFORCREWROSTERING Prob.CrewsTasksRowsColumns No.137220814121;922;626No.226320813192;343;817No.335627418554;386;271No.458432318599;412;213No.5476237122412;446;241No.6485239123715;761;156 wassolvedthroughsettingNs=2104,+=200;�=400and=0:8.Theseresultsshowedthattheoptimalsolutionwasusuallyveryclosetothatoftheprimalproblemaftersolvingtherelaxedproblem.Italsoshowedthatthealgorithmworkedwellforvarioussizesofproblemsandforvariousmodelsthattargetedondifferentcrewranks.TableIIIa,IIIbgivecomparisonsofeachoperationwiththeP-Dsubproblemsimplexalgorithms.Foreachproblem,therstcolumnindicatestheP-Dsubproblemalgorithm,andthesecondindicatesthemodiedalgorithm.Thethirdindicatesimprovedpercentagesyieldedbythemodiedalgorithmwhichequalstothedifferenceofcolumn1and2dividedbytherstcolumn.FortheP-Dsubproblemalgorithm,theparameterNswasalsosetto2104,whileanycolumnwithdj�300couldn'tjointhesubproblems.TherearesomeresultsshowninTableIIIa,IIIb.Firstly,theimprovementofthemodiedalgorithmmainlycamefromitsdramaticallydecreaseoftheiterations.Thisprovesthatthecolumnsselectioncriteriaofthemodiedalgorithmismoreeffectiveandefcient.Secondly,theimprovementoftheprobeandsortoperation(interpretedas“Probe+Sort/iter.”)alsocontributedlargelytotheoverallperformance,especiallyforthelargescaleproblems.V.CONCLUSIONInthisstudy,wehavemodiedtheconceptofprobeoperationandproposedamodiedcolumnselectioncriteriaforthecrewrosteringproblemtakingplaceinairlines,whichcanbeusedtodesignasubproblembasedalgorithm.Theideaofthemodicationistodividevariablesintogroups,andonlypricecolumnsinthemostpromisinggroupateachstep.Atthesametime,considertwofactors,oneforprimalproblemandtheotherforitsdual,toselectcolumns.Thecomputationalexperimentsonthereal-worldprob-lemshavebeenpresented,whichhaveshownthatthemod-iedalgorithmsavedaboutone-halftimewithcomparetotheprimal-dualsubproblemalgorithm. ACKNOWLEDGMENTTheauthorswouldliketothanksomelargeairlineinChinaforprovidinguswiththereal-worldproblems.Wearealsoappreciativeofthevaluablefeedbackandcommentsfromtheanonymousreferees.REFERENCES[1]CynthiaBarnhart.“AirlineScheduling:Accomplishments,Opportuni-tiesandChallenges.”LectureNotesinComputerScience,Vol.5015,2008.[2]CynthiaBarnhart,AmyCohn,EllisJohnson,DiegoKlabjan,et.al.“AirlineCrewScheduling.”RandolphW.Hall,ed.HandbookofTrans-portationScience,KluwerScienticPublishers,Boston.2003,pp.57-93.[3]NiklasKohl,StefaneE.Karisch.“AirlineCrewRostering:ProblemTypes,ModelingandOptimization.”AnnalsofOperationsResearch,Vol.127,No.1-4,2004,pp.223-257.[4]Gamache,M.,F.Soumis.“Amethodforoptimallysolvingtherosteringproblem.”G.Yu,ed.OperationsResearchintheAirlineIndustry.Kluwer,Boston,MA,pp.124-157[5]RangaAnbil,RajanTanga,EllisJohnson.“AGlobalApproachtoCrewPairingOptimization.”IBMSystemsJournal31,1991,71-78.[6]JingHu,EllisJohnson.“Computationalresultswithaprimal-dualsubproblemsimplexmethod.”OperationsResearchLetters,Vol.25,No.4,1999,pp.149-157.[7]AwantiP.Sethi,GeraldL.Thompson.“Thepivotandprobealgorithmforsolvingalinearprogram.”MathematicalProgramming,Vol.29,No.2,1984,pp.219-233.[8]FICO.“Xpress-OptimizerReferencemanual.”Release22.01.2011.