Scheduling and Project Planning

Transcription

1 My experise in inusrial problems an Projec Planning in proucion an raffic Rouing in raffic, logisics an elecommunicaion Rolf Möhring ADM III WS 5/6 DFG Research Cener MATHEON mahemaics for key echnologies P r o u k i o n s m a n a g em e n Me a ls Projecs in scheuling an logisics Areas neee for solving inusrial problems Rouing of AGVs in he Hamburg harbor r ue mp e Co cienc S Ma he Ship Traffic Opimizaion for he Kiel Canal an logisics in seel proucion Opimizing hroughpu a a airy filling line Turnaroun scheuling in chemical plans ma ics Inusrial s o rge n für ein Wu n e r w e r k. Engineering & Economics Theory Overview Par I: Sequencing an Example : Slab logisics Example : Coil coaing Example : Dairy proucion filling line Par II: uner Uncerainy Approaches o uncerainy Example : Shuown an Turnaroun Applicaions Sequencing an inpu of n iems sequence hem scheule hem w.r.. he sequence coniions may epen on subsequences cos epens on boh

2 rules an a ifferen number of shule coaers.. Formulaion Seel s are a highly iniviualize prouc, an all non-proucive in coaing epens on cerain characerisics of s, so we firs sae he mos imporan ones for a beer unersaning. Coils usually have a lengh of 5 km, ake minues o run hrough he coaing line, an some of heir cenral aribues are naurally he colors hey receive in he four coaing sages, chosen from a palee of several hunre, an heir wih, usually.8 m. We refrain from lising furher aribues, since heir lis is very long; ye all relevan informaion is inclue in our calculaions. For a concise escripion of he opimizaion ask, we shall briefly familiarize he reaer wih some coaing erminology: A coaer comprises all machinery necessary for applying one of he four layers of color o he s, primer an finish on op an boom. A hols he color currenly in use a a coaer, an each has is own rubber for applying he color o he. Naurally, a coaer has a leas one. A shule coaer has wo s, each wih is own, which can be use alernaingly o apply color. A color refers o a an filling i wih a new color. When a is use o coa a wier han is preecessor on ha, i nees o be replace, an his is referre o as a. Before enering proucion, each is unrolle an saple o he en of is preecessor. During all non-proucive, scrap s are insere in beween acual s, so essenially a never7 Coil Coaing wih Shules ening srip of shee meal is coninuously running hrough he coaing line. Afer unergoing shaping seel proucers exremely iverse proucsome porfolio:chemical The s useconiioning of heir surface, he s run hrough a op an boom primer coaer, for home appliances, for insance, alreay have heir ypical whie coaing an, a op an boom finish coaer, an hrough a secon. In he s, he respecive when bough from he seel supplier; he shee meal use for car boies alreay has an ani-corrosion coaing before i arrives a he auomoive planare fixe. Afer he coaing process, he s are rolle up again, now reay for coaing layers for pressing; s esine for builing consrucion receive heir coaings, shipping. A schemaic view of a ypical coaing line is epice in Fig.. which are very specific for echnical as well as esheic reasons, while sill a Example : Coil coaing Deails abou he scheuling phase [Höhn, König, Lübbecke, Möhring 8-9, publishe ] Tank Assignmen he seel plan. s nee o be sequence run hrough coaing line shule coaers Tankwih Assignmen primer coaer sling buffer buffer primer coaer fin assignmen wih minimum oal ile Seup work bes alpha necessaryif if Seup work necessary if Seup work necessary color s s color s color =.5 sequence alpha & largerwih wihhan hanpreecessor preecessor hashas larger has larger wih han preecessor seups in he scheuling phase Seup work necessary if Figure 5.: Some s of shee meal. They comprise up o several hunre meers of shee meal, an may weigh as much as weny ons. Figure Schemaic Seel proucers an manufacurers of coaing maerials on he one han, an isribuors of pre-coae shee meal on he oher han, have forme associaions o promoe he evoluion of coaing on naional [8] an inernaional [] levels alreay in he 96s. Progress in he evelopmen of new an improve coaing maerials an echniques fosers an ongoing iversificaion in pre-coae meal proucs, an in recen years here have been quie a few scienific publicaions on coaing, e.g., [9, 8]. Ye, o+ he j bes of our knowlege, he presen work is he firs ealing wih opimizaion in he planning process. As is ypical for pain jobs, he coaing process may be subjec o long seup s, mainly for he of equipmen, an hus very high seup cos. In orer o reuce his cos, so-calle shule coaers have been inrouce. They possess wo separae s which allow o hol wo ifferen coaings a he same, see Figure 5.. The avanage is wofol: The chemchem coaer coaer view of a coaing line wih chem, primer, an finish coaer. The chem an he boom finish coaers are sanar coaers, he remaining have shules. An insance of our opimizaion problem comprises a se [n] := {,..., n} of s o be coae. For he accoun in his paper, each j is characerize by is colors c(j) Nm on he m coaers, is wih w R, an is processing pj R+. The opimizaion goal is o minimize he makespan for coaing he se of s, i.e., he compleion of he las in he sequence. This is essenially equivalen o minimizing non-proucive, or cos, in he plan, which ensues for wo reasons: color color s s has larger Concurren Seup & Seup work necessary if Seup work necessary if fin assignmen wih fin assignmen wih minimum minimum oal ile fin assignmen wih minimum oal ile fin assignmen wih minimum oal ile Tank Assignmen finish coaer finish coaer Tank Assignmen sling Tank TankAssignmen Assignmen bes alpha sequence =.5 alpha & slingbuffer buffer sling sequence complex scheuling wih han preecessor color s color s Concurren Seup W. Ho hn, F. Ko nig, M. Lu bbecke, R. Mo hring has larger wih preecessor Concurren Seup Seup W. Ho hn, has largerhan wih han preecessor Concurren W. Ho hn, F. F. Ko nig, Ko nig, M. M. Lu bbecke, Lu bbecke, R. R. Mo hring Mo hring concurren seupseup work saves saves concurren workon on ile ile ile ile Seup work necessary if color s has larger wih han preecessor Tank Assignmen for k Shule Coaers finish coaer primer coaer finish coaer Graph moel for he scheuling phase Combining sequencing an scheuling chem coaer General problem conains k Shule Coaers: k shule coaers no parallel concurren seup Sequence generaion wih a fas geneic algorihm ur long-erm soluions. From lef o righ, opimal choicefrom of, lef wihofixe choice of erm soluions. righ, ne rule. of, wih fixe choice of l choice Auhor: Inegrae Sequencing an in Coil Coaing Managemen Science (), pp., c INFORMS Overview k shule coaers chem coaer op Special case: one shule coaer. Tank Assignmen wih k coaers.8 primer op, primer boom, primer boom, finish op, General case: k shule coaers polynomial- algorihm nee generalize inervals ue o for fixe k concurren seup ynamic programming special class of -union graphs.6 Theory Theory primer op, machine far oo slow,by increase inerval weighs savings ue oeven concurren new ieas for for seup opimum small assignmen efficien algorihm insances max weigh inepenen se in inerval graph can be solve in polynomial Pracice Pracice chem coaer boom no parallel concurren seup. Max Weigh Inep. Se in special -union graphs. base on he insighs from yn. prog. A for Tank Assignmen Opimizaion opimum assignmen srongly NP-har max weigh inepenen se in special -union graph NP-har finish op, bes alpha alpha =.5 machine finish boom 6:h :5h :h :5h 7:h 9:5h :h :5h :h :5h 6:h 8:5h :h :5h 5:h.. Figure 9 Comparison of normalize non-proucive inclue in our long-erm soluions. From lef o righ, oss are for our algorihm when using he inepenen se heurisic wih opimal choice of, wih fixe choice of =.5, an when using he online rule..6 Qualiy esing by s More expensive an longer :5h :h :5h visualizaion by Gan char, hroughpu makespan machine rivial TSP IP exper planner.6. rivial TSP. IP Airpor Berlin-Branenburg our algorihmfor represenaive shor-erm insances when rivial Figure Comparison of normalize s an makespans rivial resrice o assignmen. Bouns isplaye fromplanner lef o righ are LBriv, he sum of processing Opening elaye since June exper (human) TSP exper s, LBTSP, he sum TSP of processing s plus local cos in planner an opimal, local cos base TSP soluion, an our P LBIP. Makespans are for our soluions obaine using he online rule for scheuling,asexpece well IP o be (much) more expensive IP as for a reference soluion evise by an exper human planner where i was available. exper planner 7... Shor-Term Insances For all shor erm insances, we succeee in compuing. s by our branch-an-price approach, proving our soluions o be wihin a mos of makespan opimaliy, cf. Fig.. Ye, we i no solve all insances o ineger opimaliy an also use shor subsequences only ( 6), so he is cerainly improvable.

3 More expensive an longer... Governmen & parliamen builings in Berlin, July 999 No reay for he move from Bonn o Berlin Expece o be (much) more expensive chancellor s office by 5,, Reasons Planning an opimizaion assumes cerainy abou projec eails an aa eerminisic moels prevail Real life is subjec o many influences ha are beyon conrol machine breakowns, weaher, illness, aa eficiencies, leas o unreliable resuls an plans uneresimaion of expece makespan an cos Nee moels an echniques o cope wih uncerainy s for ealing wih uncerainy () The uneresimaion error Online opimizaion compeiive analysis, smoohe analysis Wha coul I have achieve ha I known he fuure? Sochasic ynamic opimizaion planning wih policies, minimizing in expecaion Wha can I achieve wihou knowing he fuure? Sochasic programming limis policies o recourse, uncerainy is exogenous Parially ecie now, have correcive acions laer Consier only preceence consrains Job j has ranom uraion X j wih mean x j C max (x,...,x n ) E[C max (X,...,X n )] Fulkerson 96 n X j inepenen, uniformly isribue on [,] C max (x,...,x n ) = Q{X j } = ( n Q{C max } = ) Deaile analysis of makespan isribuion Ieal: isribuion funcion F of C max 9 9 makespan Moes: perceniles 9 = inf{ Pr{C max }.9} F()

4 Shuown an Turnaroun Turnaroun Obaining sochasic informaion is har Turnaroun : resource allocaion Hagsrom 88 MEAN Given: an scheuling of large-scale Turnaroun : resourcemainenance allocaion an scheuling of large-scale aciviies in chemical manufacuring aciviies inmainenance chemical manufacuring Sochasic projec nework wih iscree inepenen processing s Flexible plan heusage scheule lengh Phase :resource! -cos raeoff problem can hire exernal workers To eermine balance urnaroun cos vs. ou of! opimal projec uraion (cos for service cos for Wane: Expece makespan! DF Given: Sochasic projec nework wih iscree inepenen processing s, resource usage vs. ou-of-service cos) esimae he risk of exceeing opimal resource usage (capaciy s, resource levelling) Phase : calculae a scheule S for Inusrial cooperaion resource leveling 6: combinaorial algorihm riskevelope analysis of S 7: a. consrains, sochasic unforeseen repairs may occuranalysis Wane: Pr{ makespan }! Projec wihin DFG Prioriy Program: Algorihm Engineering. B Opimizaion uner uncerainy in logisics an scheuling Feaures of urnaroun projecs An example: urnaroun of a cracker () They are large a single cracker has 5 jobs a urnaroun has up o 5, jobs Their preceence consrains are simple largely series-parallel wih some -lag consrains Resources are mosly workers plan workers are available a no exra cos more can be hire exernally 5 / ~ jobs, urnaroun lengh 8 ays very eaile, large variaion in processing mus respec shifs Sa.,.. So., Mo., 6.. Di., 7.. Mi., Do., LUEF.DEM.STOSS DEM.Boen 6-. DEM 7,6 Sunen.STOSS DEM.Boen -7. DEM,79 Tage KOL.TEILE ABLASSEN KT. ABL KOL.-TEILE So., TRANSP KT. TRA KOL.-TEILE SPRITZEN,S. KT. SPR KOL.-TEILE KONT.+REP KOLONNE SPRITZEN F Minuen.H. KT.KONTR KOL. SPR 6,8 Sunen IU EIGENUEBERWACHUNG KOL.-TEILE AUFZIEHEN An example: urnaroun of a cracker () Pipe fiers (Rohrschlosser), high pressure srucure of preceence consrains mosly series parallel wih shor chains 8 6 7,6 Sunen ABNAH.IU.H. S KT. AUF Tage.STOSS MON.Boen -6. MON Tage.STOSS MON.Boen 7- MON Tage high ou of service cos when lengh is exceee resource ypes, only few nee o be levele Sa.,.5. Risk is imminen inspecion may rigger repair work Fr.,.. Ohne Tiel AUERLUEFTER DEMONT MANNLOCHDECKEL MONT ML. MON DRUCKPROBE MIT EDELWASSER DP M.EDELW S. A moel for scheuling problems a se V of jobs j =,,n (wih or wihou preempion) a graph (parial orer) G of preceence consrains resource consrains, here by m ienical machines release aes rj G m=

5 Uncerainy an objecive Planning wih policies he ynamic view jobs have ranom processing s X j wih known isribuion Q j we know heir join isribuion cos funcion κ( C,,C n ) epening on he (ranom) compleion s C,,C n examples C max, w j C j, w j F j ( F j = C j r j ) plan jobs over an minimize cos Decision a (non-anicipaive) sar se S() (possibly empy) S() plan. fix enaive nex ecision plan. (eliberae ileness) nex ecision = min { plan., nex compleion } Sochasic ynamic opimizaion fixe number of jobs know join isribuion Q of processing s plan wih non-anicipaive policies Π may exploi hisory an a priory knowlege abou Q Fin policy ha minimizes / approximaes expece performance inf{e[ ] policy } performance raio of policy Π is E[ ] E[ OPT] Comparison wih online analysis vecors of processing s, each wih prob. / (,,) (,,) (,,) minimize C max possible scheules when we sar wih an Opimal in expecaion, an so raio is 7 Compeiive (average) analysis gives raios ( ) > Offline opimum is no a policy 6 E[C max ]= 7 Policies: an example Example coninue m = machines X j exp( ), inepenen Sar jobs an a = Danger: job blocks machine common ue ae penalies for laeness: v for job, w for jobs,, v << w no inerrupion of jobs mimimize E[ penalies] expensive jobs an only sequenially

6 Example coninue Example coninue Sar only job an wai for is compleion Danger: ealine is approaching sar a, fix enaive ecision if C sar an, an hen else sar a an hen an can o expensive jobs an in parallel shor span o ealine Example complee ing uncerainy for he shuown projec Expece cos for λ =, =, v =, w = = an gives he oher wo policies Jobs have iscree isribuions wih processing s early compleion : 5 normal : 7 surplus for repair : aiional surplus for geing spare pars : 5 Percenages may per group of jobs uses experience from previous urnarouns an age of equipmen Solving he urnaroun problem: Phase Uncerainy in Phase Phase : plan he scheule lengh solve a -cos raeoff problem, relax shifs an assume coninuous workers use he breakpoins on he -cos raeoff curve o calculae feasible scheules heurisically (no resource leveling) yiels alernaive rough scheules cos 55, 5, 5, 5 5 rough scheules 5 Phase : plan he scheule lengh presen he s for he rough scheules le he manager ecie manager can an see he risk Compuaion uses he sochasic risk measures on he makespan cos an risk

7 Solving he urnaroun problem: Phase Phase : calculaes a scheule S for he chosen seles neglece sie consrains such as lags levels resources heurisically uses resource flow for efining a nework N calculaes he risk base on N In aiion have compare our algorihm on small insances wih resuls from a MIP solver for a MIP formulaion Resul of he leveling algorihm Di,.9.7 Mi, Do, Fr, Sa, So, Mo,.9.7 Di, unlevele Di,.9.7 Mi, Do, Fr, Sa, So, Mo,.9.7 Di, levele