An Extension to the Tactical Planning Model for a Job Shop: Continuous-Time Control

Transcription

1 An Exension o he Tacical Planning Model for a Job Shop: Coninuous-Tie Conrol Chee Chong. Teo, Rohi Bhanagar, and Sephen C. Graves Singapore-MIT Alliance, Nanyang Technological Univ., and Massachuses Insiue Technology Absrac We develop an exension o he acical planning odel (TPM for a job shop by Graves []. The TPM is a discree-ie odel in which all ransiions occur a he sar each ie period. The ie period us be defined appropriaely in order for he odel o be eaningful. Each period us be shor enough so ha a job is unlikely o ravel hrough ore han one saion in one period. A he sae ie, he ie period needs o be long enough o jusify he assupions coninuous workflow and Markovian job oveens. We build an exension o he TPM ha overcoes his resricion period sizing by periing producion conrol over shorer ie inervals. We achieve his by deriving a coninuous-ie linear conrol rule for a single saion. We hen deerine he firs wo oens he producion level and queue lengh for he worksaion. Index Ters job shop, acical planning odel, oens producion quaniies and queue lenghs, producion soohing T I. INTRODUCTION HIS paper considers an exension o he acical planning odel a job shop by Graves []. A job shop is a process srucure in which here is a wide variey jobs and a jubled work flow hrough he shop. Due o he large variey jobs and he diverse processing requireens each job, here is no disinc workflow hrough he shop. Because he wide job variey and hus a lack prevailing work flow, producion conrol is difficul and can be very coplex. A job shop en represens he os coplex and generic for a anufacuring environen. Therefore, he abiliy o plan a job shop will provide useful insighs for producion conrol oher process srucures. Graves [] develops an analyical odel o suppor acical planning in job shops. The odel characerizes he Manuscrip received 9 h Noveber C.C. Teo is wih he Singapore-MIT Alliance (SMA a Nanyang Technological Universiy, 50 Nanyang Avenue, Singapore (eail: ps756907h@nu.edu.sg R. Bhanagar is wih he Nanyang Business School, Nanyang Technological Universiy, 50 Nanyang Avenue, Singapore (eail: ARBHATNAGAR@nu.edu.sg S. C. Graves is wih he Sloan School Manageen and he Engineering Syse Division a MIT, Cabridge MA 0239 USA. (eail: sgraves@i.edu The second and hird auhors are Fellows he Singapore-MIT Alliance. inerrelaionship variabiliy, producion soohness and work-in-process invenory in a job shop. As such, he odel provides a foundaion for he undersanding for he rade-fs inheren in he planning a job shop. The acical planning odel (TPM is a discree-ie odel in which all ransiions wihin he odel are governed by an underlying ie period. The odel assues ha all oveen jobs occurs a he sar each ie period. As such, one us se he ie period o be shor enough so ha i is unlikely for one job o ravel hrough wo successive saions in one ie period. The TPM does no explicily odel he flow discree jobs, bu raher odels he flow work due o he jobs. Work copleed in he curren period flows o downsrea sages in he nex period. However, in discree anufacuring, each job is only ransferred o he downsrea saion upon copleion. So in order o accuraely odel he job oveen, he ie period for he TPM should preferably be long o increase he fluidiy he flow he discree jobs. Furherore, he TPM assues a Markovian workflow. The validiy his assupion depends on wheher each worksaion in he shop produces a sable ix jobs. If any jobs can be copleed in one period, hen i is ore likely ha here is a sable oupu; bu his also argues for a longer ie period. In his paper we exend he odel in [] o address he odel s liiaion due o he seing he period lengh. In he nex secion, we give a lieraure review work relaed o he TPM. We hen presen a brief review he TPM in secion III. Nex, we illusrae in secion IV he liiaion he TPM due o he resricion sizing he ie periods. In secion V, we derive a new linear conrol rule ha will reove his resricion for a single-saion syse, and also deerine he firs wo oens he producion quaniy and queue lengh. We conclude in secion VI wih soe houghs on fuure research direcions. II. RELATED WORK Graves [] develops he TPM as a acical planning ool for job shop operaions. The TPM is a discree-ie linearsyse odel ha deerines he firs wo oens he producion and queue levels, given he planned lead ies he worksaions. The odel racks he workload a each saion raher han he individual jobs; he odel assues

2 ha he volue work arrivals a a saion are in fixed proporions he work copleed a upsrea saions. To dae, here are several exensions o he TPM. Parrish [2] proposes a fraework for odeling work releases o ee he delivery due dae for a finished produc. In addiion, he also shows how o adjus he conrol paraeers he TPM o change service easures in eeing deand. Graves [3] presens hree exensions o a single-saion odel he TPM. Firs, he odels a saion ha fails according o a Bernoulli process. Second, he incorporaes variabiliy due o lo-sizing, and finally, he presens he aheaical bounds on a saion wih capaciy consrain. Mihara [4] exends he work Graves [3] when he looks a an unreliable uli-saion TPM. Siilar o Graves work, he saions fail according o a Bernoulli process. Fine and Graves [5] es he TPM on a real-life job shop when hey apply he TPM o a shop ha anufacures heral conducion odules for ainfrae copuers [5]. Here, he odel is exended o allow consideraion feaures such as release policies. The odel is hen used o sudy he ipac various planning policies and he effec changes in produc ix. Hollywood [6] deonsraes how o calculae approxiaions for he seady-sae oens TPM wih general non-linear conrol rules. His odel allows for he odeling achine congesion due o capaciy loading. Oher effors adap he TPM o pull syses. Leong [7] odels a Kanban conrol syse and oher pull syses using he TPM in which work is produced a a saion whenever here is a downsrea invenory shorfall. More recenly, Graves and Hollywood develop a consaninvenory TPM in which he release work ino he shop is regulaed o ainain a consan invenory level [8]. III. REVIEW OF THE TACTICAL PLANNING MODEL The acical planning odel (TPM is a discree-ie, coninuous flow odel. All ransiions wihin he odel occur a he sar each ie period, and he jobs are odeled as workload easured in ie unis (e.g. hours. The workflow is assued o have a Markov propery: ha is, he processing requireens a a saion do no depend on how work go o he saion. As such, each individual job has no ideniy. Cenral o he TPM odel is he linear conrol rule, which is saed as Pi = ( iqi where P i is he aoun producion copleed by work saion i in ie period, Q i is he queue level a he sar period, and he paraeer i, 0 < i, is a soohing paraeer. This rule saes ha he producion P i a worksaion i is a fixed porion ( i he queue work Q i a he sar he period. In paricular, / i represens he nuber periods, on average, he work requires o ove hrough he work saion. We inerpre / i o represen he planned lead ie. We can view he conrol rule in ( as a prescripive equaion, i.e. o preserve he inegriy he planned lead ie, we us shif capaciy o heavily loaded saions. Bu ( can also be considered as a descripive equaion where producion resources are naurally flexed o accoodae he varying workloads a he saions. The queue level Q i saisfies he sandard invenory balance equaion Q + i = Qi, Pi, Ai (2 where A i is he aoun work ha arrives a worksaion i a he sar period. By subsiuing ( ino (2, we obain a firs-order soohing equaion wih i as he soohing paraeer: P i = i Pi, ( + A (3 Each worksaion can receive wo ypes arrivals; one ype arrival consiss jobs ha have heir firs processing sep a he saion, while he oher ype arrival consiss in-process jobs ha have jus copleed processing a an upsrea saion. We odel he arrivals o saion i fro anoher saion j by he equaion: Aij = φ ij Pj, + ε ij (4 A ij is he flow work arriving a saion i fro saion j a he sar period, φ ij is a posiive scalar and ε ij is a rando variable. We assue ha one uni (e.g. hour work a saion j will rigger, on average, φ ij ie unis work a saion i. The variable ε ij is a noise er ha odels uncerainy beween producion a j and arrivals o i, and is assued o be an i.i.d. rando variable wih zero ean and a known variance. The arrival o saion i is given by A = A + N (5 i j ij where N i is an i.i.d. rando variable for he workload fro new jobs ha ener he shop a saion i a ie. Subsiuing for A ij, we obain A φij Pj, + ε i, where ε i = Ni + εij (6 i = j The er ε i represens arrivals ha are no predicable fro he producion levels he previous period, and consiss work fro new jobs and noise in he flow. By assupion, he ie series ε i is independen and idenically disribued over ie. We can resae he equaions for producion (3 and for arriving work (6 in arix-vecor for: i i i j

3 P = I D + DA A P (, (7 = Φ + ε (8 where P = {P, P n }', A = {A,,A n }', and ε = {ε,,ε n }' are colun vecors rando variables, n is he nuber worksaions, I is he ideniy arix, D is a diagonal arix wih {,, n } on he diagonal, and Φ is an n-by-n arix wih eleens φ ij. By subsiuing equaion (8 ino equaion (7, we find ha = ( I D + DΦ + Dε (9 By ieraing his equaion and assuing an infinie hisory he syse, we rewrie P as an infinie series = ( I 0 s D + DΦ Dε (0 s The ean and he covariance for he noise vecor ε are denoed by µ = {µ,,µ n }', and Σ = {σ ij } respecively. The firs wo oens P are given by and where B = I D + DΦ E[ ] = ( I D + DΦ 0 = ( I Φ µ s B D 0 s Dµ ( S = var( P = Σ DB' (2 We noe ha S provides he variance he producion requireens for each saion, as well as he covariance for each pair worksaions. In addiion, we deerine he firs wo oens he queue levels. Fro (, we noe ha Q D P Therefore we have = (3 s The infinie series in equaions (, (2, (4 and (5 converge provided ha ρ(φ <, where ρ(φ denoes he specral radius Φ (see []. IV. LIMITATION OF TPM In his secion, we exaine he liiaions he TPM due o he sizing he discree ie period. This provides he oivaion for our work, which is presened in he nex secion. In he TPM, all job oveens can only occur a he sar each ie period. In order o odel oveen jobs in he acual shop, we are resriced o se he period lengh o be shor enough so ha i is highly iprobable for one job o ravel hrough ore han one saion in a single ie period. However, due o he coninuous-flow assupion, he ie period should be long relaive o he workload he individual discree jobs. This will increase he fluidiy he discree jobs and will ake he coninuous-flow assupion ore reasonable. Now we use an exaple o illusrae he discrepancies beween he odel and he acual syse due o he above conradicory objecives in period sizing. To siplify our illusraion, we consider a siple syse ha consiss wo saions in series, naely Saion i and Saion j. We furher assue ha he planned lead ie Saion i is 2 hours, while ha Saion j is hour. We se he lengh he ie period o be hour, as we assue ha i is unlikely for a job in Saion i o ravel beyond Saion j in hour. Now consider a job ha eners he epy syse and arrives a Saion i a he sar period. We suppose ha he job has a processing ie 2 hours a Saion i, and hour a Saion j. We illusrae and copare he sequence evens for he acual syse and he TPM fro period o + 2. Fig. shows he acual syse fro he sar period o + 2. The job arrives a Saion i a he sar period. Since he planned lead ie is 2 periods, he job is processed a Saion i ill he end +. The job is hen ransferred o Saion j a he sar + 2. I is hen processed a Saion j ill he end period + 2 since he planned lead ie is period. Now we look a he sae scenario in he conex he TPM. Suppose ha boh Saions i and j produce according o he TPM conrol rule in equaion (. In his case, given he planned lead ie each saion, we have i = ½ and j =. Job oveens beween he wo saions are odeled by equaion (4. We assue ha he er φ ij = 0.5 given he processing ies he job, and ε ij = 0. Fig. 2 E[ Q ] = D E[ P ] (4 and ( Q = D SD Var (5

4 Saion i processes hour work in Saion i processes hour work in + Saion j processes hour work in hours Saion i Saion j hour Saion i Saion j Saion i Saion j hour + +2 Fig. Acual syse fro sar period o hour ransferred o Saion j in sar nex period Saion i processes Saion i processes 0.5 hour work in hour work in hour ransferred o Saion j in sar nex period Saion j processes 0.5 hour work in + Saion i processes 0.25 hour work in hour ransferred o Saion j in sar nex period Saion j processes 0.25 hour work in hours Saion i Saion j hour Saion i Saion j Saion i Saion j 0.25 hour 0.5 hour 0.5 hour + +2 Fig. 2. TPM fro sar period o + 2 processed hour processed 0.5 hour 0.25 hour ie ie Acual syse TPM Fig. 3. Producion levels a Saion j in acual syse and TPM

5 illusraes he workflow in he TPM fro he sar period o + 2. As shown in he figure, Saion i processes half he in-queue workload a he sar each period ( i = ½, while Saion j processes he enire workload ( j =. And he workload processed by Saion i in each period generaes half he workload a Saion j a he sar he nex period (φ ij = 0.5. In he acual syse, he discree job oves o he downsrea saion only upon copleion. However, in he TPM, work flows as a fluid o he downsrea saion even if he discree job is no copleed. In his exaple, he discree job is spli up and oved in pars o he downsrea saion. This is due o he fac ha he period lengh is shor ( hour copared o he workload he job a Saion i (2 hours. As a resul, a workload fro he job is generaed a Saion j even hough i is sill inprocess a he upsrea saion. Fig. 3 shows he producion levels a Saion j over period o + 2 in boh he acual syse and he TPM. The producion level in he acual syse consiss a spike hour in period + 2. In he TPM, he producion is soohed, wih producion levels a 0.5 hour and 0.25 hour in period + and + 2 respecively. This exaple considers a syse only wo saions. To odel a coplex job shop using he TPM, one us se he period lengh by considering he job oveens beween all saions. On he one hand, he period lengh should be such ha i is unlikely ha a job coplees processing a ore han one work saion in a ie period. In a shop wih any saions and where jobs ove quickly beween saions, his iplies ha he period lengh be se on he order he average job workload. On he oher hand, he accuracy he TPM depends on assupions coninuous workflow. We prefer o se a long ie period relaive o he workload he jobs so ha he discree jobs will be ore fluid. Furherore, he TPM assues a Markovian workflow such ha ransiions do no depend on he hisory he syse. In essence, he odel assues ha each saion processes a relaively sable ix jobs in each ie period, so ha subsequen flow o downsrea saions is sable as well. The validiy his assupion also depends on he lengh he ie period. If only a few jobs are copleed in each period, hen i is unlikely ha here is a very sable oupu. Therefore, his assupion will be ore valid if we are able o se a longer ie period. In addiion, he resricion period sizing ay hinder he applicaion he TPM o producion planning and scheduling. In soe job shops, i akes only a shor ie (e.g. less han an hour for a job o ravel hrough ore han one saion, and hus he discree ie period has o be shor. However, he paraeers in os planning syses, such as he deand requireens, are defined in daily or weekly ie unis. Thus he abiliy o se a longer ie period will faciliae he applicaion he TPM o producion planning and scheduling. In he nex secion we describe an approach o exend he TPM o be less dependen on he choice ie period. V. MODEL In his secion, we develop a single-saion odel o overcoe he liiaions he TPM discussed in secion IV. We will derive a linear conrol rule ha accoodaes ore frequen arrivals o he work saion; as a consequence, we can now peri work o flow hrough ore han one work saion wihin a ie period. We find he firs wo oens he producion and queue lengh variables using he derived conrol rule. Wihou loss generaliy, we suppose ha each discree ie period has a lengh one ie uni. We sub-divide each ie period ino equal subinervals sub-period s, where s =, 2,,. We define = /, where is he lengh each sub-period. We assue ha work flow can arrive a he sar each sub-period. We also assue ha we se he producion in each sub-period according o he linear conrol rule (. Thus, we conrol he producion according o a finer ie grid, and we allow for ore fluid arrivals o he work saion. We resae he conrol rule ( for each sub-period s as Y(, s =.X(, s for s =, 2,, (6 where Y(, s is he producion level in sub-period s lengh, X(, s is he queue lengh a sar sub-period s lengh and is he soohing paraeer. We inerpre / as he planned lead ie; however, we now peri o assue any posiive value, and hus, we peri he planned lead ie o be less han one ie period. Equaion (6 is analogous o ( such ha he producion Y(, s in each sub-period s is a fixed fracion he queue lengh X(, s a he sar each sub-period. Now we proceed o develop he linear conrol rule for P in ers Q and A. These variables have he sae definiion as in TPM: P is he producion copleed in period, Q is he queue lengh a he sar period, and A is he arrival work o he saion in period. However, we now assue ha A does no arrive a he sar he period, bu raher arrives uniforly over period. In paricular, we assue ha in each sub-period, he arrival aoun is equal o A /. We have he following boundary condiion for he queue lengh for he firs sub-period: X(, s = = Q + A / For s >, we odel he queue lengh in he sub-period s by he sandard invenory equaion X(, s = X(, s - - Y(, s - + A / (7 By subsiuing (6 ino (7, we obain

6 X(, s = ( - X(, s - + A / (8 Now o ge an expression for P, we noe ha = P (, s = Y s (9 We su he above expressions for X(, s o find (, s = Q + ( X s + A (20 Fro (20, we observe ha s = Q + ( s + A ( s = Q + A ( (2 We now cobine (9 and (2 o ge = Q + A ( (22 Fro (22, in order o ge an expression for P, we need o find X(,. Fro (8 and repeaed subsiuion, we can hen wrie = ( + Q A ( + ( ( We can use (23 o re-wrie (22 as P = ( ( Q. ( ( A + Thus we can wrie (24 as (23 (24 γ ( = = β ( ( ( Now we proceed o deerine he coninuous-ie liis for β( and γ( as he lengh he sub-period goes o zero. This corresponds o a coninuous-ie conrol in which he producion level will saisfy (6 a every insan in ie. We use he forula li ( x 0 obain he coninuous-ie lii β( : β = li β ( = li [ ( = li [ ( ] = e For γ(, we find ha γ = li γ ( = li = li = li = ] ( ( x x = e ( ( + li [ ( ( ] ( ( β ( e = We can now resae (25 for he coninuous-ie conrol as: = β + γ (26 Q A where β and γ are given above. The balance equaion for he queue lengh for he single saion is now given by: Q. (27 = Q + A o where and P = β Q + A (25 β ( γ ( ( = ( This balance equaion differs fro (2 in he TPM, due o he new assupion ha arrivals occur coninuously hroughou a period. Hence, we define Q o be he queue lengh a he sar period, prior o any arrivals in period. By subsiuing (26 ino (27 and repeaed subsiuion, we obain: i Q = ( γ ( β A i. (28 i=

7 If we assue ha he arrivals are i.i.d. wih ean µ and variance σ 2, hen we find he wo oens for he queue lengh fro (28: E [ Q γ ] = µ = β µ 2 2 ( γ σ Var ( Q = 2 2β β Siilarly, we obain he wo oens for he producion variable: E [ P ] = µ β 2 2 Var ( P = ( γ + γ σ 2 β Thus, we have analyical expressions for he oens hese wo variables. We have expressed hese in ers he paraeers β and γ, boh which are funcions our soohing paraeer. However, i is no obvious how he variances producion and he queue lengh depend on he soohing paraeer. To provide soe insigh ino his, we graph each variance as a funcion he soohing paraeer in Fig. 4 (wih σ =. We see fro hese graphs ha he variance producion drops as we do ore soohing, i.e., sall values, or equivalenly longer planned lead ies. However, as we do ore soohing producion, boh he expeced queue lengh and is variance grow Var (P( Var (Q( Fig. 4. Variance and P and Q as a funcion for σ = These oens provide a siple way o see he fundaenal rade-fs across he hree eleens ie, capaciy and variabiliy. For given level deand variabiliy (given by he arrival process, as we reduce he planned lead ie, producion becoes ore variable and ore capaciy is required; alernaively, as we sooh 2 producion, we need less capaciy bu ore ie in ers he planned lead ie. These insighs are he sae as for he TPM. Bu he odel given here is ore general in ha we peri coninuous arrivals o he work saion and coninuous producion conrol. VI. CONCLUDING REMARKS In his paper, we exend he TPM o address is dependency on he choice ie period; in paricular, we peri work o arrive hroughou a period, as opposed o a he sar each period and we peri coninuous producion conrol. We derive a linear conrol rule by assuing ha he producion is conrolled according o a finer ie grid. We hen obain he coninuous-ie liis he conrol rule, and find he firs wo oens he producion and queue rando variables based on he conrol rule. Our nex sep is o exend he single-saion odel o a nework worksaions as would exis in a job shop. We will need o deerine how bes o odel he flow work beween saions, so as o capure acual behavior and ye sill reain analyical racabiliy. We will need o invesigae he sabiliy his odel; in paricular, we wish o find he condiions convergence for he firs wo oens producion and queue vecors. In he TPM in [], he firs wo oens converge provided ha he specral radius workflow arix Φ is less han. This iplies ha a uni work a any saion canno evenually resul in ore han one uni work for he sae saion. I is ineres o evaluae he convergence condiions his odel as copared o he TPM in []. In addiion, we have no esablished he benefis his exension over he original TPM under differen job shop condiions. I is worhwhile o carry ou a sudy, perhaps hrough a copuaional experien, o copare how his exension perfors relaive o he TPM in differen operaing condiions, such as he sabiliy job ix and speed job oveen beween saions. Anoher opporuniy for fuure research is o exend his odel o incorporae congesion effecs due o capaciy loading. This would involve relaxing he linear conrol assupions for each saion. Hollywood [6] has developed an approxiaion for he TPM wih nonlinear conrol rules. I ay be possible o inegrae his approxiaion ino his odel. REFERENCES [] S.C. Graves, A Tacical Planning Model for a job shop, Operaions Research, 34, 986, pp [2] S.H. Parrish, Exensions o a odel for acical planning in a job shop Environen, S.M. Thesis, Operaions Research Cener, Massachuses Insiue Technology, June 987. [3] S.C. Graves, Exensions o a Tacical Planning Model for a job shop, Proceedings he 27 h IEEE Conference on Decision and Conrol, Ausin, Texas, Deceber 988.

8 [4] S. Mihara, A Tacical Planning Model for a job shop wih unreliable work saions and capaciy consrains, S.M. Thesis, Operaions Research Cener, MIT, Cabridge MA, January 988. [5] C. H. Fine and S. C. Graves, A Tacical Planning Model for anufacuring subcoponens ainfrae copuers, J. Mfg. Oper. Mg., 2, 989, pp [6] Hollywood, John S An Approxiae Planning Model for a Job Shop wih Nonlinear Producion Rules, working paper, 32 pp. [7] T.Y. Leong, A Tacical Planning Model for a ixed push and pull syse, Ph.D. progra second year paper, Sloan School Manageen, Massachuses Insiue Technology, July 987. [8] S. C. Graves and J. S. Hollywood, A consan-invenory Tacical Planning Model for a job shop, Working paper, January 200, revised March 2004, 35 pp.