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从提供的文件信息来看，这篇文章的内容涉及到了生产调度、机器故障、调度预测性以及调度方法的探讨。下面将详细展开这些知识点。标题中的“Predictable scheduling of a single machine with breakdowns and sensitive jobs”指的是对单一机器进行的可预测调度，该调度需要处理机器故障和敏感作业（即对时间要求较高的作业）。这里的“敏感”可以理解为对作业完成时间有着严格要求的作业，这些作业的延误可能会对整个生产计划和客户满意度产生较大影响。描述中提到了预测性生产调度的两个重要功能：它需要优化车间绩效，即将车间资源分配给不同的作业，以提高某种车间绩效指标（比如成本、时间或质量）。预测性调度作为基础，支持物料采购、预防性维护以及向外部或内部客户交付订单等规划活动。这个过程强调的是调度不仅要能预测未来的作业计划，而且要能在实际生产中遇到机器故障等干扰时，能够快速适应并最小化对计划作业的影响，同时还要尽可能维持高车间绩效。在标签中提到的“国外文献”表明这篇文章很可能是一篇发表于国际期刊的专业论文，这通常意味着其内容经过了同行评审，并具有一定的研究深度和学术价值。从部分内容中我们可以了解到，文章发表在《***PUTERINTEGRATEDMANUFACTURING》期刊，1999年，第12卷，第1期，页码为15到38。文章由Sanjay V. Mehta和Reha Uzsoy撰写。他们提出了一个预测调度方法，其中预测性调度是在构建时就考虑了处理干扰的目标。具体来说，文章讨论的调度模型是针对单一机器，在动态作业到达和随机机器故障的环境中，最小化最大延迟（Lmax）的问题。文章提出的关键点包括： 1. 机器故障时如何修改调度以减少对计划作业的影响。 2. 调度修改过程中可能造成的活动执行延迟或不可行。 3. 在不影响计划活动的前提下，维持高车间绩效的调度模型。 4. 通过减少计划完成时间与实际完成时间的差异来衡量干扰对计划活动的影响。 5. 研究表明，预测性调度提供高度的可预测性，而且在实际调度绩效上只有小幅度的牺牲。从这些信息可以了解到，在制造业中，尤其在高技术行业如半导体制造中，生产调度是极为关键的环节。由于这些行业通常面对着大型复杂的制造系统，简单的手工技术很难获得良好的调度结果。因此，有效的生产调度程序对于在今天的全球竞争市场中实现及时且成本效益高的生产至关重要。此外，文章中还提到了调度过程通常是这样的：基于当前订单的当前状态信息，为将来的一定时间段（如一天或一个班次）开发一个预测性调度计划（Sp），然后将此计划释放到车间。但在这个过程中，预测性调度在执行中会因为机器故障等干扰的发生而被修改，而这种修改可能会造成基于预测性调度计划的活动执行延迟或变得不可行。这篇文章不仅为我们提供了一个预测性调度模型的构建方法，同时也探讨了在实际生产过程中如何处理突发情况，如机器故障，以及如何在保证车间绩效的前提下，实现调度活动的高可预测性和鲁棒性。这为生产管理提供了重要的理论支持，并对于实际应用有较高的指导意义。

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Pre dictable sche duling of a single m achine

subj e ct to bre akdowns

SANJAY V. MEHTA and REHA UZSOY

Abstract. Th e pre dictive p roduction sche dule has two im -

por tant function s; allocating shop re sou rces to the differe nt

jobs to optim ize som e m easure of sh op pe rform an ce and

se rving as a basis for plan ning activities such as mate rial

pro cu rem e nt, preven tive mainte nance and de livery of orde rs

to e xternal or inte rn al cus tom ers. This sche dule is modified

durin g execution on the occ urre n ce of disrup tions such as

m achine breakdowns. Th e sche dule modification proce ss ma y

de lay or rende r in feasible the e xecution of activities plan ned

on the basis of the pre dictive sche dule . Th us it is o f interest to

de ve lop pre dicti ve sche dule s which can absorb disruptions

without affecting plann ed ac tivities while main tainin g high

shop perform ance . A pre dictable sch eduling approach is

presen te d, whe re the pre dictive sche du le is bu ilt with such

obje ctives. The effects of disruptions on planne d activities are

m easure d by the differe nce be twee n plan ne d and realized job

com pletion time s. Th e spe cific sche duling m od el considere d is

m inim izin g m axim um laten e ss ( Lmax) on a s ingle m ach i n e

with dynam ic job arrivals and random m achine bre akdown s. It

is shown that pre dictable sch eduling provide s high pre dict-

abilit y with m inor sacrifices in re alized sche dule pe rform an ce .

1. Intro d u c tio n

Tim e ly and cost e ffective production is be com in g

in cre asingly im portan t in today’ s global com pe titive

m arke ts. Prod uction sch e d ulin g plays an im portan t role

in in d u cin g such fac tor y be h avio u r by e fficie n tly

allocatin g scarce re sources to com pe ting activitie s over

tim e . Effe ctive production sch e dulin g proce du re s are

particularly im p ortan t in th e large , com ple x m an ufac-

turin g system s e ncou n te re d in high -techn ology indus-

trie s such as se m icon du ctor m an ufacturin g, wh e re

sim ple m an ual tech n ique s are u nlike ly to yie ld good

re sults ( Uzsoy et al. 199 2, 1994) .

In m an y production syste m s th e sche dulin g process

is as follows. A pre dictive sch edule S

is de velop e d for a

ce rtain pe riod of time in t o the future , such as a day or a

sh ift, b ase d on th e state of th e sh o p floor an d

in form ation on cu rre n t ord e rs. Th e sc h e d u le is

re le ase d to the shop floor at th e start of the pe riod.

Wh en unforese e n di sruption s such as m ach in e bre ak-

down s occu r on the sh op floor, S

is m odifie d to

m ain ta in fe asibility or im prove perform an ce . We shall

refe r to th e ac tivity of sc h e d u le m odificatio n as

re sch e du lin g. At the en d of the plan n in g h orizon , we

h ave the re alize d sch e dule S

th at was actually e xe cute d

on th e sh op floor, which m ay d iffe r substan tially from S

du e to re sch edulin g. Figu re 1 sh ows th is proce ss, wh ich

we shall refe r to as Predictive

Reactive Sche du lin g.

Th e pre dictive sch e dule S

se rves two im portan t

fun cti on s. Th e first is to allocate shop resou rces to the

d iffe re n t jobs to optim ize som e m e asure of sh op

pe rform an ce. Th e se cond , as poin te d out by Wu et al.

( 1993 b) , is to se rve as a basis for plan n i n g activities such

as m ate rial procure m e n t, pre ve n tive m ain te nan ce an d

de live ry of orde rs to e xtern a l or in te rnal custom e rs.

Resch e d ulin g on th e shop floor aim e d at im proving

sh op pe rform an ce m ay delay or re nd e r infe asible th e

e xe cution of th e se activit ie s as plan n e d. Fre que n t

re sch e du lin g can al so m ake sh op be haviour h ard to

p re dict, reduc in g th e e ffe ctive n e ss of h igh e r le ve l

production plan n in g system s. Th us it is of in te re st to

d e ve lop p red ictive sch e d u le s wh ich m in im ize th e

e ffe cts of d isrup tion s on plan n e d ac tivitie s wh ile

m ain ta in in g high sh op pe rform an ce . We shall re fer to

sch e d ulin g with such dual objective s as Predictable

Schedu lin g ( PS) . In th is p aper we an alyse a predictable

sch e dulin g proble m an d de ve lop solution te chnique s.

Unde rstan din g th e nature a n d fre que n cy of th e

disruption s e n countered on th e shop floor is e sse n tial

to e ffe ctive schedu lin g in the face of un ce rtain ty. McKay

et al. ( 1989) ide ntify three type s of un ce rtaintie s in

m an ufac turin g e n viron m e n ts; com ple te un known s,

suspicion s abou t th e future a n d known unce rtain tie s.

Comple te un kn own s are tota lly unpredictable e ven ts

such as a sudde n strike about which n o advan ce

INT. J. CO MPUTER INTEGRATED MANUFACTURING

, 1999,

VOL

. 12,

. 1, 15 ± 38

0951-192 X

99 $12.00

1999 Taylor & Francis Ltd

Authors: S.V. Me hta, R. Uzsoy, Sc h ool of Indu strial En gin e e rin g, 1287 Grissom

Hall, Purdue Unive rsity, We st Lafaye tte , Indian a 47907-12 87, USA.

in formation is availab le . Little can be don e abou t the se

e x c e p t re ac t. H e n c e d e ve lopm e n t o f p r e dictable

sch e dules for syste m s with this type of unce rta in ty is

difficult. Suspicion s about th e future arise from t h e

in tuition an d e xpe rience of the h um an sche dule r.

Th e se are hard to in corporate i n to sch e duling algo-

rith m s as th e y ar e d iffic u lt to qu an tify. Kn o w n

un ce rtain ties are th ose a bout wh ich som e in form at ion

is availab le wh e n th e pre d i ctive sch e dule is ge n e rated,

such as m achine bre ak d owns wh ose frequ e n cy an d

duration m ay be ch aracte rize d by probability distribu-

tion s. Schedulin g algorith m s can a d dre ss som e kn own

un ce rtain ties by con siderin g th is type of in form ation

while de ve lopin g th e pre d ictive sche dule . It is th is type

of u n cert ain ty th at we focus on in th is pape r.

We consider a PS ap proach wh e re available infor-

m ation regardin g kn own un ce rtain tie s is u se d to

ge ne rate a pre dictive sche dule S

. Th e effe cts of

disruptions is m e asure d by th e de viation of th e re alize d

job com ple tion tim e ( in the re alize d sch e d ule ) from its

plan n e d com ple tion tim e ( in th e pre dictive sch edule ) .

Th e se deviations are re duce d by inse rtin g additional

idle tim e in to S

base d on th e se que n ce of jobs and the

n atu re an d fre qu e n cy of the disruption s that are

expe cte d to occur. Th us th e com ple t ion tim e s of jobs

in th e pre d ictive sche dule are funct ion s of th e j ob

se quen ce an d the am oun t of addition al idle tim e

in se rte d. Th e se com p letion time s are the n u se d to

quote de live ry date s to custom e rs or to plan activitie s

such as se tup s or pe rson n el sch ed ule s. Th e actual

sch e dule re alize d on the sh op floor will diffe r from S

it is re vise d in th e fac e of disrup tio n s an d as

un nece ssary idle time is absorbe d.

Th e specific sche dulin g mod e l we consider is that of

m in im izing m axim u m late n e ss ( Lm ax) on a sing le

m ach in e with dynam ic job arrivals an d ran d om break-

down s. All data e xcep t bre a kd own time s an d duration s

are de te rm in istic an d kn own wh e n ge n e ratin g th e

predictive sche dule. Tim e be tween bre akd owns an d

tim e to re p air distribution s are assum e d to be availab le .

Exte n sive com putation al e xp e rim e n ts show that it is

possible to d eve lop sch edu le s with h igh predictability

with very m in or sacrif ice s in Lm ax( S

) . We also show

th at PS p e rform s we ll e ve n whe n th e breakdown

distribution param e te rs can n ot be e stim ate d ac curate ly.

O ur reason s for focusin g on this m ode l are se ve ral.

Firstly, stud yin g pre dictable sche dulin g u sing a single

m ach ine m ode l provid e s in sigh ts that can be use d to

exte n d the ap proach to m ore com plex, multimachin e

m ultistage s yste m s. Se condly, m an y re al world produc-

tion syste m s are ch aracte rize d by th e p re se n ce of a

bott len e ck sta ge wh ose pe rform an ce de te rm ine s to a

large e xte n t th e perform an ce of th e en tire shop . Th e

de ve lopm e nt of an e f fective sche duli n g proced ure f or

this m ach in e is ofte n a significan t step toward s a viable

ap proach to sche du ling the e ntire facility. Th is is

in dee d the ap proa ch ad vocate d by Gold ratt ( 1986) in

his Th e ory of Con s traints, particularly h is Drum-Buffe r-

Rope approach to shop floor con trol. Fin ally, th is

particular single m ach in e proble m studie d he re form s

an im portan t com p on e nt of dec om position he uristics

for m ore com ple x proble m s. In the se decom p osition

approac he s, such as th e Sh iftin g Bottle ne ck ap proach

of Adam s et al. ( 1 988) or the various d ecom p os ition

procedure s discusse d by O vacik and Uzsoy ( 1997) , a

m ultim achine m ul tistage sch e dulin g proble m is de com -

pose d into a num be r of single machine subp roble m s.

Th e subproble m s are solve d in dividually i n orde r of

the ir criticality to th e solution of the overall proble m .

O vacik an d Uzsoy ( 1997) pre sen t e xte n sive com puta-

tional e xp e rim en ts sh owin g that th e se m e th ods sig-

n ifican tly ou t-pe rform dispatch in g ru le s in se ve ral

in dustrial environ m e n ts, particularly th ose in se m icon -

ductor m an ufacturing.

In th e n e xt section , we re vie w pre vious re late d work

on sch e duling in th e face of unce rtain tie s. In s e ction 3

we form ulate th e pre dictable sche du lin g problem for a

sin gle m achin e with ran dom bre akdown s, an d discuss a

spe cial case . We the n add re ss the proble m of de te rm in -

in g th e optim al am oun t of idle tim e to in se rt in th e

pr e dictive sch e d u le . Howe ve r, th is is d ifficu lt to

op tim ize e xplicitly d ue to the com ple x, stoch astic

nature of th e proble m . Th us in section 4 we deve lop

a num be r of surrogate m e asure s of sch e dule pre dict-

ability which can be calc ulate d easily for a give n

predictive sche dule an d provide a good e stim ate of

sch e dule pre d ictability. Th e ir perform an ce is eval u ate d

by com putation al e xpe rim e n ts in se ction 5. Using t h e

be st of th ese s u rrog ate m e asures, we d e ve lop various PS

he uristics in se ction 6. Th e se h e uristics are e valu ate d

through exte nsive com putat ional e xperim e n tation in

se ction 7. Results show that th e PS he uristics provide

S.V. Mehta an d R. Uzsoy

Figure 1. Pre d ictive

re active sche dulin g proce ss

h ig h p re d ictability with ve ry little d e grad ation in

Lm ax( S

) com pare d to PRS. In se ction 8, we show that

PS outpe rform s PRS e ve n wh e n bre akd own p aram e te rs

can n ot be e stim ate d accurate ly. We conclude t h e pape r

with a sum m ary an d a discu ssion of future direction s.

2. Pre vio u s re late d wo rk

Previous approach es to sche du ling in th e pre se n ce

of un ce rtain tie s can be classifie d into fou r m ain

cate gorie s as sh own in figure 2. No past ap proach h as

explicitly con side re d pre dictability wh ile bu ilding pre -

dictive sche du le s. We focus on s u ch an ap proac h in th is

pap e r.

2.1. Completely reactive approaches

In com ple tely reactive approaches, no firm sche du le

is ge ne rat e d an d de cisions are m ad e locally in re al tim e .

Th is approach fails to provid e an y plan for othe r

activit ies, a n d it is hard to pre dict syste m pe rform an ce

as de cision s are m ad e locally in re al tim e . A freque ntly

use d reactive ap proach is p riority dispatchin g rule s,

whe re wh ene ve r th e m ach in e be com e s free the avail-

able job with high e st priority is se le ct e d for proce ssing.

Th e priority of a job is de te rm in ed base d on job an d

m ach in e attribu te s. An e xte n sive discussion of local

pri ority dispatch in g rule s can be foun d in Bh a skaran

an d Pin e do ( 1991) .

2.2. Predictiv e

Reactive Schedu lin g (PRS)

In th is approach , wh ich i s ill u strated in figu re 1, a

predictive sche dule is ge n e rat e d in advan ce with th e

obje ctive of optim izin g s h op p e rform an c e with out

considerin g possible disruption s on the shop floor.

Whe n disruptions occur durin g e xe cution, the pre -

dictive sch e dule is m odifie d ( resch e dule d ) to m ain tain

fe asibility or im prove pe rform an ce .

Th e m ain issu es in this ap proach are wh e th e r to

pe rform re sch e dulin g at a particular disruption a n d

wh at re sche d uling m e th od to u se . In con tin u ous

re sch e d u lin g we re sch e dule e ve ry tim e an e ve n t

alte rin g th e state of the system occurs ( Ram an et al.

1989 ) . In periodic resch e dulin g, re sch edulin g take s

place only at re gu l ar in te rvals in tim e . Contin uous

re sch e dulin g provid e s be tter realize d sche du le p e rfor-

m an ce but has h igh e r com putation al burde n an d lowe r

predictability than pe riodic re sche dulin g ( Adam an d

Surkis 1980, Ch urch an d Uzsoy 199 2, Muhlem an n et al.

1982 ) . Event drive n re sch eduling com bine s t h e ad van -

tag e s of con tinuous an d p e riodic resch e du lin g, basing

th e de cision wh e th er to resche du le on the st ate of the

syste m at the tim e of th e d isruption . Re sche du lin g is

also p e rf orm ed at re gu lar in te rvals. Yam am oto a n d Nof

( 1985 ) an d Ch urch an d Uzsoy ( 1992) sh ow th at e ven t-

drive n re sch e du lin g can provide com parable p e rfor-

m an ce to con tin uous re sch eduling with lowe r com p uta-

tional burd e n an d h igh er pre dictability. H owe ve r,

con tinuous re sch e duling m ay be re quire d for certain

types of disruption s such as m ach in e breakdown s to

m ain ta in feasibility.

O n ce th e tim in g of the re sch e du lin g is e stablishe d,

we n eed to de te rm in e h ow to re sch e du le . Existing

re sch e du lin g m e thods can be classifi e d as righ t-sh ift

re sch e du lin g, com plete re sch eduling, m atch -up re sch e -

du ling an d m u lti-obje ctive re sch eduling. In righ t-shift

re sch e du lin g, the sch e dule is m oved forward in tim e to

th e e n d of th e disruption while the original sequ e n ce of

ope ration s is main tain e d. If possible , idle time be twee n

th e op eration s in th e pre d ictive sch e dule is absorbe d

( Sadeh et al. 1993) . In complete re sch e dulin g, all

op e rations in the syste m afte r the disrup tion are

re sch e du le d. Th is provides bette r realize d sche dule

pe rform an ce an d lowe r predictability th an righ t-shift

re sch e du lin g. Bean et al. ( 199 1) propose a m atch-up

re sch e du lin g ap proach whe re the m odifie d sch e dule

m atches th e pre dictive sch e du le at s om e point in th e

fu tu re . The ir e xp e rim e n tal re su lts sh ow th at th is

m e thod provide s near-opt im al solution s wh ile ach ie ving

h igh e r p re d ictability than com ple te re sch e d u lin g .

Multiobje ctive re s ch e duling ap proache s con side r both

sch e dule pe rform an ce an d pre dictability during resch e -

du ling. Wu et al. ( 19 93b) prop os e such an ap proach for

a single m achine proble m with dynam ic job arrivals an d

de live ry ti m e s. Th e y associate de lay costs, e arline ss cost

an d se quence chan ge costs with th e ch an ge s in th e

Predictable schedu lin g to machin e breakdown s

Figure 2. Past app roache s to sche duling in the pre sence of

un ce rtain tie s

predictive sche dule due to re sch e dulin g an d form an

obje ctive fun ction which is a we ighte d sum of m ake span

( Cm ax) an d total m odification cost. Th e ir re sults show

that th is ap proach im prove s pre d ictability wh ile re tain -

in g ne ar optim al Cm ax. Th is is a re active ap proach as

the pre dictability issue is on ly con side re d wh ile re sch e -

duling . In this p ape r, we use sim ilar cost m e asure s but

con side r predictability wh ile buildin g th e p re d ictive

sch e dule.

2.3. Robu st schedulin g approaches

Robu st sche dulin g ap proach e s focus on building

the pre dictive sch e dule to m in im ize t h e effe cts of

disruptions on th e p e rform an ce m e asure valu e of th e

re alize d sch e d ule . It diffe rs from our pre dictable

sch e dulin g ap proach that trie s to e n sure that th e

pr e d ictive an d re alize d sc h e dule s do n o t d iffe r

dra stically in te rm s of job com pletion times, while

m ain tain in g high re alize d sch e d ule pe rform an c e .

Le on et a l. ( 1990) c onside r a robust jo b sh op

sch e dulin g proble m with m achin e bre akdown s, pro-

cessing tim e var iations an d minim izin g Cmax a n d

Cm ax d eviation s as pe rform an ce m e asu re s. Wu et al.

( 1993 a) propose a de com position ap proa ch to ach ie ve

sc h e d u le robu stne ss. T h e y conside r a job sh op

proble m with proce ssing tim e variation s an d m inim iz-

in g total we igh te d tardin ess as obje ctive f un ction . For a

discussion of robust sche du lin g, th e re ade r is refe rre d

to Le on et al. ( 199 0) , Wu et al. ( 199 3a) , an d Dan ie ls

an d Kouvelis ( 1995) .

2.4. Kn owledge based approaches

Kn owle d ge base d approach es provid e a m e ch an ism

for se le ctin g an ap propriate re s ch e duling policy from

am on g a n um be r of alte rnative s. Such syste m s also allow

m an ual in te rve n tion by th e h um an sch e dule r.

Sm ith ( 1992 ) discusse s th e con strain t dire cte d

re sch e dulin g cap a bilitie s in O PIS ( O pportunistic In-

te llige n t Sch e dule r) , a kn owle dge base d sch e dulin g

syste m d e ve lo pe d at Car n e gie Me llon Un ive rsity.

Furth er discussion of O PIS can be found in Sm ith

( 1992 ) , Smith ( 198 7) an d Sm ith et al. ( 199 0) . Sadeh et

al. ( 1993) discu ss th e con s traint prop ag ation base d

re sch e dulin g approach use d in Mi cro-Boss, a fin ite -

capacity production sch e d ulin g and con trol syste m

de velope d at Carn e gie Me llon Un iversity. McKay et al.

( 1989 ) discu ss WATPASS, a finite -loadin g job sh op

sch e dulin g syste m deve lop e d by Wate rloo Man age m e n t

of In te grate d Man u facturing Syste m s Re se arch Group.

Resch e dulin g is pe rform e d inte ractive ly by the sche du-

le r. Dutta ( 199 0) propose s an s -con trol m e ch an ism

m on itorin g the en vironm e n t an d takin g co rre ctive

action s to diffe ren t disruption s. Bap tiste an d Favre l

( 1993) discuss schem e s to re prese n t alternative sch e -

dule s while de ve lopin g a p re dic tive sch e dule . Whe n

re sch e dulin g is n e ce ssary, an alt e rnative sche dule from

the availab le se t is sele cte d an d e xe cu ted. An e xte n sive

discussion of knowle dge -base d re active sch e duling can

be foun d in Szelke an d Ke rr ( 1994 ) .

In sum m ary, n o past approach has addre sse d th e

predictability issue while bu i ldin g th e predic t ive sch e -

dule . In th e n e xt se ction , we formulate the single

m ach ine pre dic table sche dulin g proble m wh e re th e

effe cts of sche dule m odification s on plan n e d activitie s is

considere d wh ile buildin g th e predic tive sche du le .

3. Pro b le m f o rm u latio n an d an alysis

In th is pap e r we foc u s on th e p roble m of

m in im izin g Lm ax on a s in gle m ach in e with dyn am ic

job arrivals an d ran dom breakdown s. Th is proble m

with out bre akdown s h as be en extensive ly studie d an d

shown to be stron gly NP-hard ( Garey an d Joh n son

1979) . Th e Earlie st Due Dat e ( EDD) dispatch in g ru le

provid e s an op tim al solution to th e static sin gle

m ach ine proble m . Lawle r’ s algo rithm can be use d

for the static sin gle m ach in e problem with pre cede nc e

constrain ts ( Pin e do 1995) . Bake r an d Su ( 197 4) an d

Carlie r ( 1982) pre se n t bran ch an d boun d algorith m s,

while Potts ( 198 0) , Carlie r ( 1982 ) an d Hall an d

Sh m oys ( 1992 ) an alyse he u ristics for th e sin gle

m ach in e proble m with dynam ic job arrivals. This

re se arch provid e s a stron g foundation for the proble m

un der conside ration .

Conside r a set N of jobs to be processe d on a

m ach ine . Th e proce ssin g time s p

, re le ase tim e s r

, an d

due d ate s d

for e ach job i

N are de te rm in istic an d

known a priori. Th e m achine is subje ct to ran dom

bre akdown s. We assum e th at the m ach in e bre akdown

durations an d occurre n ce s can be de scri be d using

probability distribution s. This in f orm ation is often

availab le in practice from m ain t e n an ce re cords. Con -

side r a pre d ic tive sch e d u le S

ge n e rate d at th e

be gin n in g of the p lan n ing h orizon. Le t C

( S

) be th e

com ple tion tim e of job i in th is sche dule . S

is e xe cute d

on the shop floor an d revise d using som e re sch e duling

m e thod when bre akdown s occur. At the e n d of th e

plan n ing horizon , we have a re alize d s ch e dule S

. Le t

( S

) be the com p le tion tim e of job i in S

. Le t

Lm ax ( S

) an d Lm ax( S

) be th e maxim um late ness

valu es of S

an d S

respectively. We in cur th e followin g

costs d ue to jobs not be in g com plete d at their plan ne d

com ple tion tim e s:

S.V. Mehta an d R. Uzsoy

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