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毕业中英文翻译

英译

*********学院

*****UNIVER S IT Y

汉)

设计题目:设计一座年产550 万吨良坯的转炉车间,产品以板坯为主

学生姓名:**

学号:

专业班级:09 冶金*班

学部:材料化工部

指导教师:赵** 讲师

2013 年05 月31 日

原文

Productionschedulinginasteelmaking-continuouscasting plant

Abstract

Inthispaperwe describean optimizationprocedureforplanningthe productionofsteelingotsinasteelmakingcontinuous casting plant.The strictrequirements ofthe production process defeated most of the earlier approaches to steelmakingcontinuous casting production scheduling, mainly due to the lack ofinformation intheoptimizationmodels. Our formulation oftheproblem is basedonthe alternative graph,which is a generalizationof the disjunctive graph ofRoy andSussman.Thealternativegraphformulation allow us to describe in detail all theconstraintsthatarerelevantfortheschedulingproblem.Wethensolvetheproblem byusingabeamsearchprocedure,andcompare our results with a lower bound of theoptimal solutions and with the actual performance obtained in the plant.Computational experienceshow stheeffectivenessofthisapproach.

Keywords:Scheduling;Productionplanning;Steelmaking1.Introduction:

Steel industry is veryrichinterms of operational problems that can be modeledandsolvedbyusingcomputerizedtechniques.Infact,duetothecompetitivepressure,manyinternatio nalironandsteelcorporationsaremovingto the just-in-time (JIT)productionconcepts. Common features of the JIT manufacturing systems are thesmall size ofthelotsandthe requests for high quality products, timing delivery,increasedproductivityandreducedcosts.

Toalarge extent, in the steel industry, scheduling and related issues are stillcarried out by human schedulers whose computer support mainly consists of asimulatoroftheplant,whichisusedtoevaluateandcomparedifferentscenarios.Ontheotherhand,manycompa nies arenowdevelopingandimplementingcomputerizedsystemsfor addressing such operational problems, as reflected in the professionalmeetings on the subject. These systems help to increase productivity and timing

delivery, while reducing energy consumption andproductioncosts.Steel schedulingis known tobeoneof the mostdifficultindustrial scheduling problems. Researchersand practitioners have approached the related scheduling problems using theformulations and tool sof various disciplines,mostly artificial intelligence Dorn etal,1996,DornandShams,1996,TürksenandFazelZarandi,1998 andSantosetal,2003and mathematical programming Dutta and Fourer,2001,Naphade et al,2001, Tang etal,2000,Moon and Hrymak,1999, Harjunkoski and Grossmann,2001 and Tang etal,2002.

Inthispaperwedeal in particularwiththe problem ofschedulingtheproductionofstainlesssteelingotsina

productionlinelocatedincentralItaly.Theplantproducesalargevarietyofingots,tobeusedinawidesetofapplic ations,characterized

bythesizeandchemicalcomposition,or“g rade”.T heingotformationprocess,orsteelmakingcontinuous casting process, has extremely strict requirements ofmaterial continuityandflowtimetofulfillinordertoachievesuitablepropertiesonthefinalproduct,thusmakingtheprod uctionscheduling problem particularly challenging.Show the layoutoftheproductionline.Theproductionisorganizedinlots,eachlotbeingcomposedofagivennumberofla dlestobecastconsecutively.Theladlesbelonging tothesamelotareidentical and are associated to a specific production order. The productionscheduling problem consists of determining the scheduling of operations to beperformedonmoltensteelatthedifferentproductionstages from steelmaking tocontinuous casting. In particular, the processing of stainless steel consists of asequenceofhigh temperatureoperationsstartingwiththeloading ofscrap ironinanelectric arc furnace (EAF). The time to melt the scrap iron ranges from 70 to80 minutes,includinga few minutesforsetupandfurnacemaintenanceoperations.

The liquid steel is poured into ladles that a crane transports to a subsequentmachine,called argon oxygen decarburization unit(AOD),wherenickel, chromiumand other elements are added to the steel in order to meet the chemical qualityrequirements. Usually the durationofthe operationon the AOD ranges between70and80minutes.Between EAF and AOD thereis room for storing up to three ladlesbu t, si n c e the store d ste e l c oo l s dow n, i t m u st be re hea ted i n the A OD . A fter the AOD

theladlesaretransportedtoa ladle furnace(LF) which can hostat mosttwo ladles.Evenif some operationsare executedintheLF (nomorethan30 minutes),in practiceitactsas a buffer to maintain the ladles at the proper temperature before the lastoperation,tobeexecutedinthecontinuouscaster(CC).BetweentheLF and the CCthereisa bufferthatcan holdatmostone ladle.A ladle can stay inthebufferatmost10minutes,otherwisethe liquid steel could cool down.IntheCC the liquid steel iscastandcooled toform slabs,thetime required forcasting oneladle ranges from60to70minutes.MoreovertheCC needstobetooledwitha particulartool,calledflyingtundish,whichdeterminesthe formatofslabs.Iftwo subsequentladles belongtothesame lot,they must be cast without interruption, otherwise the tundish deteriorates,and a setup is necessary in order to substitute it. However,the tundish must bechanged anyway when switching from a lot to another, as well as after a givennumberof ladles, ranging from three to seven, which depends onthe steel quality.The size ofa lot may vary,therefore,from oneto seven ladles.The setup time needsabout60minutes,duringwhichtheCCisblocked.Theproblem is to schedule theproductionofagivennumberoflotswiththeobjectiveofminimizing thecompletiontime ofthe lastscheduled lot,i.e.minimizingthe makespan.Wereferto this problemassteelmakingcontinuous casting (SCC) problem. The time horizonfor a planningperiodis1 week, which corresponds to an average of 120 ladles and30 lots foratypicalreal sizeproblem.Fig.2 shows theGanttchartofa schedule inwhichtwo lotsareproduced.Thefirstlotiscomposedofthreeladles,andthesecondiscomposed

oftwoladles.Theprocessingtimesare not realistic,but are useful to highlight theproductionrequirementsofa feasible schedule.

Summarizing the above discussion, the SCC problem can be modeled as apermutationflowshopwithlimitedbuffersandthefollowingadditionalconstraints:

•Allpartsina lotmustbesequencedconsecutively.

•Therearesequenceindependentsetuptimesonthelastmachine(CC)betweentw oconsecutivelots.

•Thereisano-waitconstraintbetweenanytwo consecutivecastingoperationsinthe sa m e lot.

Thereisaperishability constraintinthebufferbeforethelastmachine(CC).

Thepaperisorganizedasfollows.In Section2 we briefly review the literaturerelatedtotheS CCproblem,inSection3weintroducethenotationand,inSection4,wediscusstwomodelsforthe SCC problem. InSection 5,we describesome proceduresforcomputinglower bounds to the optimal solution,In Section 6 we describe ourbeam search procedure for the SCC problem and,in Section 7,we illustrate thecomputational experience.ConclusionsfollowinSection8.

2.L iteraturereview

Inthissectionwebrieflyreview theliteraturerelated tothispaper.Inparticular,in the first part of this section we discuss the literature on problems arising insteelmakingplants.Inthesecondpartwereviewsomegeneralapproachesforsolvingscheduling problems that can be applied to industrial cases, including the SCCproblemas a special case. In the last part ofthis section we briefly review somerelevantpapersdealingwiththebeamsearchtechnique,analgorithmic procedurethatw e u se i n thi s pa per to sol v e the S C C proble m.

2.1Scheduling problem insteelmaking plants

Inan extensive survey paper,Dutta andFourer(2001) discuss several problemsarisinginasteelmakingplant.Theproblemsrangefrom productmixandblendingtoproduction scheduling and cutting stock. For all these problems, mathematicalprogrammingapproachesarepresented.Inviewoftheirextensive survey,we provideonly a bri e f su m m a ry o f re c e nt w ork s he re .

Naphade et al.(2001)formulate the scheduling problem in a steelmaking plantusinga mixed integer formulation andsolve itusing a two level heuristic approach.Tangetal.(2000) formulatetheSCC problem usinga non-linearprogrammingmodel,andthenconvertitintoalinearprogrammingmodel, solvableusingstandardsoftwarepackages. In another recent work of Tang et al. (2002), the SCC problem isformulated by using an integer programming model and it is solved combiningLagrangianrelaxation,dynamicprogrammingandheuristics.

HarjunkoskiandGrossmann (2001)considertheschedulingproblems,arising inthe produ ction of ste e l , for a l i ne c o m posed o f tw o pa ra l l el EA F s, a n A OD , a n L F a nd

a CC. They develop a decomposition strategy, which is able to reduce thecomputationeffortdrastically. Thisstrategy allows totackle industrial-sizeproblems,based on 1 week time horizon and80 ladles, obtaining feasible solutionswhicharealwayswithin3%froma lower boundofthe optimum. Themain layout differencesbetween our settingand thesettingconsidered in Harjunkoski andGrossmann(2001)arethatinourproblemtheladlesaregroupedinlotswhereasin Harjunkoski andGrossmanneachladleisindependentoftheothers,and anotherdifference is due tothe pre se nc e of tw o pa ra l l e l EA Fs in their produ cti on l i ne. Other rel eva nt w ork s focu son the opti m i za tio n of the C C m a c hi ne S a ntos e t a l ., 2 0 0 3 a nd S c h w i ndt a ndTrautmann,2003 orfurnaces,coolersandcraneoperations(Moon&Hrymak,1999).

2.2Machinescheduling withcomplicating constraints

Adifferentstream ofresearch related to SCC problem deals, more in general,withothermachineschedulingproblemsexhibitingthesame complicatingconstraintsoftheSCC problem. Forexample,scheduling problems withlimitedcapacity buffersariseintheproductionofconcretewares(Grabowski,Pempera,&Smutnicki,1997),chemicalproductsR eklaitis,1982 andKimetal.,2000,aswellasintheschedulingoftrains (Şa hin, 1999) and in the flow control of packet switching communicationnetworks (Arbib,Italiano,&Panconesi, 1990).Inthis context,three differentsettingsaretypicallydistinguished

(see,forexampleSchwindt&Trautmann,2002)unlimitedintermediate storage(UIS), finite intermediate storage (FIS) and non-intermediatestorage (NIS). Hall and Sriskandarajah (1996) model the absence of intermediatebuffers(NIS) asablocking constraint.Inthis casea job,havingcompletedprocessingona machine,remainsonituntilthenextmachinebecomesavailable forprocessing.

Atwomachine flowshopschedulingproblem,in which the capacity of theintermediatebufferislimited and the jobshave to be produced in lotsof identicalparts,isstudiedbyAgnetis,PacciarelliandRossi(1997).Theproblem isprovedtobe- ha rd , a nd a n a lg ori thm i s proposed for i ts solu ti o n. T he a lg orithm ru ns inpolynomialtimeanditis exact if some conditions on the batch sizes are met,otherwise an approximation result holds in general. Pranzo (2004) extends these

resultstothemoregeneralcasewithsequenceindependentsetup and removal timesfor the ba tche s.

McCormick,Pinedo,ShenkerandWolf(1989)study a flow shop schedulingproblemin an assembly line having finitecapacity buffersbetween machines (FIS).They model the positions of the intermediate buffers as machines with zeroprocessing time. The problem with limited buffers can be therefore studied as ablockingproblemwhereallmachines have nointermediate buffers.They alsoshowtha t, on c e a se qu e nc e for the j obs ha s be e n fou nd, the s ta rti ng ti m e of the j obs on a l lmachines canbeeasilycomputed ona precedence constraintsgraph.They distinguishtwocategoriesofarcs:processingarcs,havinglengthequaltotheprocessing time oftheassociatedoperations,anddummyarcs,ofzerolength,representingtheblockingconstra ints.

Sanma rtí, Friedler and Puigjaner (1998) study the production scheduling ofamultipurposebatch plant.They introduce the schedule-graphrepresentationwhich isabletorepresent both unlimited intermediate storage (UIS) and non-intermediatestorage(NIS)situations.Oncea completeschedule graphis produced,themakespanonthegraph provides the timing of each operation.They implementa branch andboundmethodinwhichthelowerbound on the makespan is evaluated ona partialschedulegraph.Computationalexperience shows that theiralgorithm outperforms astandard mixed integer programming solver. Romero, Puigjaner, Holczinger andFriedler(2004) extendthe schedule-graphrepresentation,orS-graph,andcombine itwith a feasibility check based on linear programming to deal with differentintermediate storage policy, including no-wait (or zero wait, ZW) and finiteintermediatestorage (FIS). Intheirbranch and bound procedurethelower boundisbasedonthelongestpathcomputationundertheassumptionofunlimitedwaitingtime.Ifthisisnotthecas e,e.g. when no-waitoccurs,thelowerbound onthemakespan iscalculatedusingalinearprogrammingmodel.

Otherrelevantworksariseinthemanagementofperishableitems.A commodityissaidtobeperishable if some ofits characteristics are subjectto deteriorationwithrespectto customer/producerrequirements.Thecooling ofliquid steel is thereforea

form of perishing. The perishability issue is approached in various ways in theliteratureonscheduling, butmostofthe schedulingmodelsinthiscontextsufferfroma lack of information. For example, an approximation often introduced whenschedulingperishablegoodswithhighdecayrate,is theintroductionoftightno-waitconstraintsGrabowskietal.,1997,HallandSriskandarajah,1996andPinedo,1995.Othermodels Mc Cormick et al., 1989 S.T. Mc Cormick, M.L. Pinedo,S. ShenkerandB.W olf,Sequencinginanassemblylinewithblockingtominimizecycletime,OperationsResearch37(1989)(6),pp.925–935.FullTextviaCrossRefViewRecordinScopusCitedByinScopus(81)McCormicketal.,1989

andSanma rtíetal.,1998donotexplicitlyconsiderperishabilityconstraints,thus providing only a lowerboundonthemakespanofa sequencewhenperishabilityoccurs.

Mascis and Pacciarelli (2002) introduce the alternative graph model, whichextendsthedisjunctive graph formulation of R oy and Sussman (1964) in order torepresent blocking and perishability constraints. In particular, the alternative graphgeneralizes the precedence constraints graph (Mc Cormick et al., 1989) and theS-graphRomeroetal.,2004and Sanma rtí et al., 1998 to include perishabilityconstraints.ThemainadvantageofthealternativegraphwithrespecttotheS-graphisthatno-waitandperishability constraintscan berepresenteddirectlywithinthegraph,without the need for additional feasibility check procedures. This compactrepresentationalsoenables the computation of more effective lower bounds, whichcan be used to accelerate optimization procedures. Pacciarelli(2002) formulates theSCC problem with the alternative graph and solves it with a simple and effectivegreedy heuristic.Inthis paper,we improve boththe formulationoftheSCC problemandthesolutionalgorithmpresentedinPacciarelli(2002).In particular,we presentamorecompactformulation,basedonthealternativegraph,andabeamsearchsolutionprocedure.

2.3Thebeam searchtechnique

Thebeamsearchtechniqueisaheuristicsearchstrategy, firstused in artificialintelligenceforthespeechrecognition problem (Lowerre, 1976), and then used byFox(1983) forsolvingcomplexschedulingproblems.Itconsistsofa truncatedbranch

andbound,inwhichthenodes ofthebranchtreearevisitedinbreadthfirstorder,andonlyth e β m ostpromising nodes at each levelareselected as nodes tobranch from.Theparameterβis calledthebeamwidth,andlarger values ofβ u sually result inincreasedcomputingtimeandquality ofthesolutions.Thenodeevaluationprocessatea c h l e v el i s the k e y i ssu e of a ny bea m sea rc h, w here a com pro m i se betw e e n solu ti onquality and computing time must be found. A common approach consists ofcomputinganupperboundanda lowerboundassociatedwiththecandidatenode,andusing these values to evaluate th e β mo st promising nodes. The beam searchprocedure has been applied tothe UIS job shopproblem by Sabuncuoglu andBayiz(1999)usingdispatchingrules,andbyWernerandWinkler(1995)usinganinsertionheuristicandalocal search procedure.The beam search procedure proposedin thispaperisbasedonthesameheuristicusedinPacciarelli(2002).

In thi s se cti on w e d e scri be the nota tion w e u se throu g hou t the pa per a ndthealternativegraphformulationofMascisandPacciarelli(2002).Intheusualdefinitionofajob shopschedulingproblem,ajobmustbeprocessedonasetofmachines,thesequenceof machines for each job is prescribed, and the processing of a job on amachinerequires a fixed, non-preemptive, processing time. The problem consistsofallocatingmachinesto competing jobsover time, subjectto the constraintthat eachmachinecan handleatmostonejobata time.Inparticular,theschedulingproblem isa flow shopproblem ifthesequence ofmachines visited by eachjobis thesame fora l l the jobs.

T he proce ss ing o f a j ob on a m a chine i s ca l l ed a n opera tion, a nd there fore ea chjobconsistsofasequence ofoperationsthathavetobe processedina specifiedorder.T he g oa l i s to m i ni m i z e the com pl etion ti m e o f the l a st o pera tion.

In thi s pa per w e fo cu s on the se qu e nci ng o f op era ti ons ra ther tha n jobs. W e ha vetherefore a set of operations {o0,o1,…,on} which have to be performed on mmachines {m1,m2,…,mm}. Each operation oi requires a specified amount ofprocessingpionaspecifiedmachineM(i),andcannotbeinterruptedfromits startingtim e ti to i ts com pl etio n ti m e ti+pi . o0 a nd o n a re du m m y op era ti ons, w i th zero

processingtime,thatwecall“start”and“fi ni sh”,respectively. Each machinecan processonly oneoperationatatime.

Thereis a setofprecedence relations among operations.A precedence relation(i,j)isa constraint on the starting time of operation oj, with respect to ti. Moreprec i se l y , the s ta rti ng ti m e s o f the su c c e sso r oj m u st b e g rea ter or equ a l to the s ta rti ngtimeofthe predecessor oi plus a given delay dij, which in our model can beeitherpositive,nullornegative.A positive delay dij may represent, for example, the factthatoperationojmaystartprocessingonlyafterthecompletionofitspredecessoroi.Inthis casedij=pi,i.e.tj≥ti+pi. A delay smallerorequal tozerorepresents a synchronization between thestartingtimesofthetwo operations.Finally, we assumethato0precedeso1,…,on,andonfollowso0,…,on−1.

Precedencerelationsaredividedinto

twosets:fixedandalternative.Alternativeprecedencerelationsarepartitionedintopairs.They usually representthe constraintsthateachmachinecanprocessonly oneoperationata time.

A schedule is an assignment of starting times t0,t1,…,tn to operationso0,o1,…,onrespectively, suchthatallfixed precedence relations,andexactlyoneforeach pair of the alternative precedence relations, are satisfied. Without loss ofg e nera l i ty w e a ssu m e t0 =0 . T he g oa l i s to m in im i ze the s ta rti ng ti m e of opera ti o n on.Thisproblemcanbetherefore formulatedasaparticulardisjunctiveprogram(Balas,1979),i.e.alinearprogramwith“e xclusive or”()conditi ons,calleddisjunctions.

Probl e m1

Associatinganodetoeachoperation,Problem 1 can beusefully representedbythe tri ple thatwe call alternativegraph.Thealternative graphisas

follows.ThereisasetofnodesNassociatedtotheoperations,asetofdirectedarcsF anda setof pairs of directed arcs A associated to precedence relations between operations.Arcs in thesetF arefixed and dijis thelength ofarc (i,j) F.Arcs in thesetA arealternative.If((i,j),(h,k)) A, we say that(i,j) and(h,k) arepairedandthat(i,j) isthe

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