A Study of Moment Recursion Models for Tactical Planning of a Job Shop: Literature Survey and Research Opportunities

Transcription

1 A Sudy of Momen Recursion Models for Tacical Planning of a Job Shop: Lieraure Survey and Research Opporuniies Chee Chong Teo Singapore-MT Alliance, N2-B2C-15, Nanyang Technological Universiy, 50, Nanyang Ave, Singapore Absrac The Momen Recursion (MR) models are a class of models for acical planning of job shops or oher processing neworks. The MR model can be used o deermine or approximae he firs wo momens of producion quaniies and queue lenghs a each work saion of a job shop. Knowledge of hese wo momens is sufficien o carry ou a variey of performance evaluaion, opimizaion and decision-suppor applicaions. This paper presens a lieraure survey of he Momen-Recursion models. Limiaions in he exising research and possible research opporuniies are also discussed. Based on he research opporuniies discussed, we are in he process of building a model ha aemps o fill hese research gaps. ndex Terms job shop acical planning model, lieraure survey and research opporuniies in momen recursion models, momens of producion quaniies and queue lenghs. T. NTRODUCTON HS paper considers a class of models for acical planning of job shops or oher processing neworks. This class of models is called he Momen Recursion (MR) models. A job shop is a process srucure in which here is a wide variey of jobs and a jumbled work flow hrough he shop. Due o he large variey of jobs and he diverse processing requiremens of each job, here is no disinc workflow hrough he shop. Specifically, a work saion may receive jobs from differen saions and jobs compleed a he saion may be roued o one of he several saions or may leave he shop if compleed. Because of he wide variey of jobs and hus a lack of prevailing work flow, producion conrol is difficul and can be very complex. xamples of job shops are he manufacuring of cusom-made producs, such as machining workshops, where each job has unique requiremens from differen cusomers. Manuscrip received 3 rd November C.C. Teo is a PhD suden wih he Singapore-MT Alliance (SMA) under he nnovaions in Manufacuring Sysems and Technology (MST) program hosed a Nanyang Technological Universiy, Singapore. ( ps h@nu.edu.sg). A job shop ofen represens he mos complex and generic form of a manufacuring environmen. Therefore, he abiliy o plan a job shop will provide useful insighs for producion conrol of such oher process srucures. By acical planning, we imply ha we are no concerned wih he deailed scheduling issues. Here, we are more ineresed in idenifying he dominan flows in he job shop and subsequenly modeling he job shop for acical planning purposes, such as capaciy planning. We consider a class of models called he MR Models for modeling and analyzing job shops. The defining feaures of a MR model are: 1) is a discree ime model, such ha work is compleed during fixed-lengh periods, and work arrivals and ransfers occur a he sar of hese periods. 2) ach saion of he nework produces a quaniy ha depends on he work-in-process level a ha saion hrough some producion funcion. 3) Workflows are Markovian such ha processing requiremens do no depend on how he work ges o he saion. 4) Work arrivals are sochasic and are characerized by finie mean and variance. 5) Recursion equaions can be wrien o describe he relaionship beween he firs wo momens of producion quaniies and queue lenghs in periods. The MR models can be used o deermine or approximae he firs wo momens of producion quaniies and queue lenghs a each work saion of a job shop. Knowledge of he firs wo momens is sufficien o carry ou a variey of performance evaluaion, opimizaion and decision-suppor applicaions. For example, we may use MR models o help opimize processing neworks of job shops. We may opimize he performance of a job shop (e.g. hroughpu rae or work-inprocess invenory) wih a budge consrain for capaciy, or minimize he capaciy cos wih a performance arge (hroughpu rae, work-in-process or lead ime). One of he main capabiliies of he MR models is is abiliy o compue he second momen of producion volume, which is one of he key measures of variabiliy in

2 a manufacuring faciliy. Variabiliy of producion volume is imporan for several reasons. A larger degree of variabiliy implies a higher level of finished-goods invenory for a make-o-order manufacurer, or a longer lead-ime quoed o cusomers in a make-o-order environmen. n addiion, variabiliy can upse producion planning and schedules. Overime, use of subconracors, emporary workers and oher expediing services are ofen associaed wih high variabiliy. This paper presens a lieraure survey of he Momen- Recursion models which is useful for modeling and analyzing job shops. Limiaions in he exising research and some possible research opporuniies are discussed. is hoped ha such a discussion would provide direcions for fuure work o make MR models more realisic and applicable o real-life manufacuring environmen. The remainder of he paper is organized ino hree secions. Secion gives a review of he concep of a MR model. Secion presens he lieraure survey of MR models as well as oher relaed research. Secion V discusses he limiaions and possible research opporuniies in his area.. CONCPT OF A MOMNT-RCURSON MODL n his secion, we presen he basic concep of a MR model. There are a number of variaions wihin his class of models, largely in he conrol rules and release policies. The differen variaions will be discussed in he nex secion. Here, we will only look a he MR models in is mos general form. Consider a job shop or oher nework of processing saions. The shop is modeled in discree ime, such ha he saions process some amoun of work-inprocess during each ime period, and ransfer he work o oher saions or ou of he nework a he sar of he nex ime period. The workflow is modeled as work hours, raher han as a se of disinc jobs. ach saion i mus saisfy he following elemenary invenory balance equaion, where Q i, Q i, = Qi, 1 Pi, 1 + Ai, (1) is he work-in-process level (boh in queue and in service) a he sar of period, work processed in period -1, and P i, 1 A i, is he amoun of is he amoun of work ha eners saion i a he sar of period. We wrie (1) for all saions in marix form as Q + = Q 1 P 1 A (2) where is a vecor of work-in-process, is a vecor Q P 1 A of producion quaniies, and P is a vecor of arrivals. The following assumpions are made abou P and A. is a funcion of he work-in-process a he sar of period, i.e. P = f ( Q ), where f is a producion funcion or conrol rule ha relaes he producion quaniies of a saion in a period wih he work-in-process a he sar of he same period. has wo componens. The firs A componen A () represens work ha is ransferred inernally wihin he nework beween saions and is a deerminisic funcion of P, i.e. A P ). corresponds o he concep ha compleed work a one saion in one period riggers work a oher saions in he nex period. The second componen consiss of work ha arrives o saions exernally from ouside he shop. A can be independen and idenically disribued, or can be a funcion of Q i.e. A Q ) which allows he 1 modeling of pull release policies such as consaninvenory job shops. By subsiuing he above expressions for and 1 ( 1 A ( 1 P 1 ino (2), we obain he recursion equaion (3) ha ha forms he basis of he MR models. i = Qi, 1 Pi, 1( Q 1) + A [ P 1( Q 1 Q, )] + A (3) f we know he expeced value and variance of A, we can express (3) in erms of and, or alernaively and Q Q 1 P P 1 A. By repeaedly ieraing he resuling closeform expressions, and assuming ha he job shop has a defined seady sae, we can obain a converging infinie series. We can deermine he expeced values and he variance of his series o obain he firs wo momens of P and Q.. RLATD LTRATUR We devoe his secion o a lieraure survey of MR models as well as oher relaed models of processing neworks. A. Momen-Recursion Models The firs paper on he MR models is by Graves in which he developed he Tacical Planning Model (TPM) [1]. The saions in his model use a linear conrol rule = Q, where α i is a producion smoohing P i, α i i, parameer for saion i and is inverse 1/ α i is he planned lead ime. The conrol rule implies ha he producion rae a saion i is a fixed proporion α of is queue lengh in each ime period, and is consisen wih he assignmen of a planned lead ime o each saion. The longer he planned lead ime, he greaer amoun of producion smoohing he saion will be subjeced o. This conrol rule is based on he approximae analyical model for producion smoohing developed by Cruickshanks, Drescher and Graves [2]. By using an example of a facory ha produces grinding i

3 machines, he illusraed how o use he model o evaluae he choice of he planned lead ime and also o find a good specificaion ha will resul in an accepable shop behavior. Parrish presened some exensions o he TPM [3]. Firs, he proposed a framework for modeling work releases o mee a delivery due dae for a finished produc. n addiion, he showed how o apply he TPM modeling framework o generae wo service measures - he probabiliy ha demand exceeds invenory and he average number of successive periods in which demand is no me. n addiion, he also showed how o adjus he conrol parameers of he TPM o change hese service measures. Leong modeled he Kanban and oher pull sysems using he TPM [4]. n a pull sysem, work is produced a a saion whenever here is a downsream invenory shorfall. Here, he linear conrol rule is P = α T Q ), where T i is i, i ( i i, he arge invenory level. Graves presened a model o provide a rough-cu assessmen of boh he saffing and componen invenory levels of a repair depo [5]. The repair rae of he depo has a producion funcion ha resembles he linear conrol rule of he TPM. He firs suggesed he piecewise linear producion funcion P = min[ Q, K + Q / n] where K and n are consans. The raionale behind his funcion is ha he oal producion level canno exceed he work-inprocess (backlog of failed unis in his case). Here, he producion level is se equal o he sum of a consan erm K and a erm ha is proporional o he work-in-process Q/n. However, his funcion canno be evaluaed direcly. Hence, he suggesed an approximaion P = K + Q / n. Here he values of K and n are se so ha he probabiliy of he producion quaniies P exceeding he work-in-process Q is small. Graves presened hree exensions o a single saion model of he TPM [6]. Firs, he modeled a saion ha fails according o a Bernoulli process and he duraion of each failure is exacly one period Second, he derived he approximae seady sae momens for he TPM wih losizing. n his model, work compleed by he saion is merged ino los of fixed size, and he los are roued probabilisically. Finally, he presened he mahemaical bounds on he behavior of a saion wih a bounded conrol rule which depics he capaciy consrain a he saion. The conrol rule is of he form P = min( αq, M ), where M is he capaciy consrain of he saion. Mihara exended he work of Graves [6] on unreliable single saion in TPM when he looked a unreliable mulisaion TPM [7]. Bu similar o Graves work, he saions also fail according o a Bernoulli process. He also performed simulaion sudies of a muli-saion TPM in which each saion i uses bounded conrol rules of he form P = min( αq, M ) as discussed in [6]. He found ha he behavior of he bounded models approaches he behavior of he unbounded TPM provided ha he capaciy consrain of each saion is sufficienly large relaive o he workload. Fine and Graves applied he TPM o a real-world job shop ha manufacured hermal conducion modules for BM mainframes [8]. Here, he model was exended o allow consideraion of feaures such as release policies. By using regression mehods, he parameers were fied o he observed daa and hey found some empirical evidence for he use of linear conrol rules in pracice. The model was hen used o sudy he impac of various planning policies and he effec of changes in produc mix. Hollywood made several exensions o he TPM [9]. He defined he class of MR models of which he feaures are saed in Secion 1. Previously, work on his class of model had been limied o he TPM and direc exensions o he TPM. Consequenly, all research assumed ha he producion is a fixed fracion of work-in-process, and ha work arrivals are saionary fluid arrivals. He showed ha he TPM is one model in a much larger class of models ha may be analyzed hrough similar echniques. He expanded he TPM o include models wih general linear conrol rules which are also known as affine conrol rules. These rules allow producion o be a weighed sum of invenory levels a muliple saions, plus a random noise erm. He hen applied hese models on a nework ha uses highly sophisicaed affine conrol rules, i.e. he proporional resoraion rules of Denardo and Tang [10]. Nex, he demonsraed how o calculae approximaions for he seady-sae momens of MR models wih general non-linear conrol rules. These models allow for he modeling of a wide range of realisic machine and human behavior, including machine congesion and effecs of overime work. He suggesed a non-linear conrol rule which is based on Karmarkar s clearing funcion [11]. However, he resuls are approximaions. rrors can be large if he nework is heavily loaded and work arrivals are highly variable. Hollywood also showed how o se up and solve opimizaion problems relaed o MR models, including maximum performance (minimum queuing imes or queue lenghs, and maximum hroughpu) and minimum cos problem. n doing so, he showed how o model capaciy requiremens for models wih linear or general conrol rules. He also used he underlying recursion equaions o find he ransien behavior of MR models ha are subjeced o changes in he nework. For simple nework changes, he found he analyical expression for he ransiens, exending he work of Parrish [3]. Using his ransien analysis, he also found ha he opimal conrol rules are linear funcions of he work-in-process, hereby jusifying he use of linear conrol rules. Graves and Hollywood developed a consan-invenory TPM in which he release of work ino he shop is regulaed o mainain a consan invenory level [12]. They were able o deermine he firs wo momens for he producion random vecor for such a release policy and also characerized he condiions for which he producion levels converge o a seady sae. They hen illusraed he

4 use of he model wih an applicaion and showed he benefis of such a release policy wih a compuaional experimen. B. Oher Relaed Models We now compare he MR models wih oher relaed models. Our focus here is o compare he modeling framework and he approaches, raher han he inen of he models. Queuing models are widely used o model processing neworks. Jackson developed he basic queuing model for open queuing neworks, now known as he Jackson neworks [13][14]. Similar o he MR models, he deailed sequencing of jobs is no considered and work flows can be characerized in a complex job shop. Gordon and Newell exended Jackson s work o a closed queuing sysem in which he number of jobs remains consan [15]. There is now a large lieraure on queuing neworks ha exends and generalizes Jackson s work, wih much of i validaing agains simulaion sudies. Buzaco and Yao, and Suri and Sanders provided an exensive survey on queuing models [16][17]. Queuing models usually assume he arrival and service imes o be independen and idenically disribued. n order o compue he exac soluions, he arrivals are assumed o follow a Poisson process wih exponenially disribued service imes. Whi provided approximaions for general neworks wih G/G/m queues using he firs wo momens of iner-arrival and service imes [18][19]. The resuling queuing model is known as he Queuing Nework Analyzer. Biran and Tirupai exended his model by improving he approximaion accuracy of he muliple cusomer class version of he model [20]. They achieved his by developing an improved approach o esimae he ineracion beween cusomer classes, albei i requires more complex compuaion. Much recenly, here is some lieraure ha focuses on he approximaions of heavy-raffic queuing models. Work in his area includes Harrison and Williams, Harrison and Wein [21]-[23]. These are Brownian moion approximaions based on he fac ha deparures from heavily-loaded saions are approximaely exponenially disribued. However, he major drawback of hese models is ha he accuracy of hese models largely depends on he raffic-inensiies of he queues, and hus hey are more applicable o heavily-loaded queues. MR models allow he modeling of splis and merges in workflows, i.e. a single job can spli ino muliple jobs, or muliple jobs can merge ino a single job upon compleion and moves o downsream saions. Bu his is generally no possible in queuing models. Furhermore, MR models allows he compuaion of firs wo momens of boh producion quaniies and queue lenghs, while queuing models usually gives only he firs momens. However, i is difficul o model probabilisic job rouing in MR models which is a basic feaure of queuing models, alhough Graves presened an approximaion mehod for such modeling for he TPM [6]. Generally, MR models are more suiable if he producion oupu per ime period can be expressed as a funcion of he oal work and he nework has well-defined workflows. Queuing models are more appropriae if he nework has saions wih independen and idenically disribued service imes and has disinc classes of cusomers. Besides queuing models, i is also worhwhile o compare he MR models wih deerminisic planning models ha are ofen used for aggregae and capaciy planning. Compared o MR models which are sochasic models, he deerminisic models assume he modeling parameers o be deerminisic. The expeced producion quaniies and queue lenghs are usually assumed o be deerminisic quaniies. The main advanage of deerminisic models over MR models is ha i is possible o model complicaed producion rules such as bounded conrol rules which is no possible wih MR models. However, capaciy is usually a hard consrain in deerminisic models. A saion i processes up o is fixed capaciy in each discree ime period, i.e. producion of saion i is given by he bounded conrol rule P i = min(q i, M i ). As a resul, such models do no accoun for he lead ime or work-in-process consequences of capaciy loading, as lead ime of producion (and work-in-process) is consan regardless of he amoun of capaciy loading. Reviews of hese models can be found in Hax, Baker, and Biran and Tirupai [24]- [26]. One deerminisic model ha is closely relaed o he MR models is he npu/oupu conrol by Wigh (1970). The npu/oupu conrol is a way of analyzing he consequences of order release decisions of maerial planning. Similar o he MR models, i is a discree-ime processing nework model a a shop level. n his model, work arriving a each saion is deermined and he amoun of work processed is compued by he bounded conrol rule. Work in excess of capaciy is carried over o he nex period. The discree reamen of work order and job sep of his model causes i o resemble a discree-ime simulaion. Thus he resuling compuaions required are as deailed and complex as he real simulaion, and hence i is difficul o embed i in opimizaion procedures. Karmarkar s deerminisic model akes ino accoun of he capaciy-loading effec on he lead ime and work-inprocess by a clearing funcion [11]. Similar o he conrol rules in MR models, he clearing funcion describes he amoun of oupu cleared from he manufacuring faciliy as a funcion of is work-in-process. The funcion is MQ P = (4) β + Q where P is he producion rae, Q is he work-in-process level, M is he capaciy level, and β is a parameer ha deermines he clearing rae. He adaped (4) for a discree period dynamic planning model ha direcly models workin-process and finished invenories. Since i is a deerminisic model, i can be incorporaed readily in mahemaical programming echniques.

5 Unlike he clearing funcion, he linear conrol rule in he TPM does no capure he effec of capaciy loading on work-in-process and producion lead imes. Hence Hollywood suggesed a non-linear conrol rule based on he clearing funcion and also illusraed an approximaion mehod o compue he seady-sae momens of MR models wih general non-linear conrol rules [9]. n his secion, we have presened he lieraure on MR models as well as oher relaed models. However, here sill exis some significan research gaps, which will be discussed in he nex secion. V. RSARCH OPPORTUNTS There is subsanial lieraure on MR models as presened in he previous secion. However, addiional work is needed o make MR models more realisic and applicable in real-life manufacuring faciliies. There are more unresolved issues han can be lised; we will look a some ha seem ineresing and valuable. Many producion faciliies show congesion effec due o capaciy loading. Hollywood has derived he non-linear conrol rule ha is based on Karmarkar s clearing funcion which is a concave funcion designed o model sauraion and congesion behavior [9][11]. Bu Hollywood s non-linear conrol rule does no perform well in high raffic-inensiy condiions and when he arrival is highly variable. This grealy limis he use of his model since saions of high raffic-inensiy (boleneck saions) are usually he ones ha are more crucial, and hus require a more horough performance evaluaion. Hollywood performed a discree-ime simulaion sudy o validae his approximaions. He found ha errors in approximaing he second momens of boh producion and queue lenghs of a single-sage saion are larger han 50% when subjeced o a raffic-inensiy of 0.9 and work arrivals wih a coefficien of variabiliy equal o 1.0. Therefore, more effor is required o improve he non-linear conrol rules or he approximaion mehod. One feaure of he MR models is ha work ransfers beween saions occur only a he beginning of each discree ime period. n many manufacuring faciliies, jobs move o he nex processing saion immediaely afer compleion a each saion. Thus here is a coninuous shule of jobs beween saions of he nework. Graves suggesed ha he MR models sill applies o such an environmen if he periods are carefully sized [1]. The periods mus be long enough such ha a significan amoun of work is done in each period, bu shor enough ha a job is unlikely o ravel hrough more han one saion in a paricular period. However, such period-sizing migh be resricive in general applicaions. n many producion sysems, i akes only a shor ime (e.g. less han half an hour) for a job o ravel hrough more han one saion, and hus he discree ime period has o be shor. n mos applicaions, a longer ime period migh be more valuable. This is because he MR model oupus, such as he producion variance, are more meaningful if hey are defined over a longer ime. is usually more useful for a producion manager o know he producion variance of a shif, raher han over a onehour period. For example, a manager planning he capaciy migh use he expression Capaciy = [ P ]+ kσ, where [ P ] is he expeced producion quaniies per period, σp is he sandard deviaion of producion quaniies, and k is he safey facor ha convers he sandard deviaion of he hroughpu ino performance guaranee. k equals o he z-value of sandard normal disribuion if P is normally disribued. Here, i is hard o se he value of k if σ is of a one-hour period even if P is normally disribued, as i is difficul o deermine he amoun of performance guaranee needed for each one-hour period. A performance guaranee for a shif would be more meaningful. is generally complex o deermine he producion variance of a longer ime period given he producion variance of shorer ime periods. This is because he producion quaniies of successive periods are generally correlaed. Hollywood s approximaion for non-linear conrol rules is more accurae if he arrival process is less variable [9]. n many oher forms of nework approximaions in which here is a sochasic arrival process, such as Whi s approximaion of G/G/m queues [18][19], a high level of arrival variabiliy generally worsens he accuracy of he approximaion. By having a longer discree ime period, he variabiliy of he amoun of work ha arrives in each period will be lower due o variabiliy pooling over a longer period of ime. Therefore, in he case where an exac soluion of he MR model canno be obained and hus requires an approximaion, a longer discree ime period will probably improve he accuracy of he approximaion. We are in he process of building a model ha aemps o fill he above menioned research gaps. This model accouns for machine congesion and has significanly more accurae second momen approximaions han Hollywood s non-linear conrol rule model. Furhermore, i models coninuous flow of jobs and can have discree ime periods longer han he ime for a job o ravel hrough more han one saion. Simulaion sudies were carried ou o validae he accuracy of he model. Preliminary resuls of he sudies are encouraging. As menioned in Secion, one of he main capabiliies of he MR model is is abiliy o deermine or approximae he variances of producion volume. Machine failure is he main source of unplanned variabiliy in he producion floor. Hence, he abiliy o include machine failures in he MR model will grealy improve is applicabiliy. Graves modeled an unreliable single saion in which he saion fails wih a fixed probabiliy p and he duraion of he failure is exacly one period [6]. This is of limied realism compared o acual machine failures and addiional effor is herefore required in his aspec of modeling. P p

6 RFRNCS [1] S.C. Graves, A Tacical Planning Model for a job shop, Operaions Research, 34, 1986, pp [2] A.B. Cruickshanks, R. D. Drescher and S. C. Graves, A sudy of producion smoohing in a job shop environmen, Managemen Science, 30, 1984, [3] S.H. Parrish, xensions o a model for acical planning in a job shop nvironmen, S.M. Thesis, Operaions Research Cener, Massachuses nsiue of Technology, June [4] T.Y. Leong, A Tacical Planning Model for a mixed push and pull sysem, Ph.D. program second year paper, Sloan School of Managemen, Massachuses nsiue of Technology, July [5] S.C. Graves, Deermining he spares and saffing levels for a repair depo, J. Mfg. Oper. Mg, 1, 1988, pp [6] S.C. Graves, xensions o a Tacical Planning Model for a job shop, Proceedings of he 27 h Conference on Decision and Conrol, Ausin, Texas, December [7] S. Mihara, A Tacical Planning Model for a job shop wih unreliable work saions and capaciy consrains, S.M. Thesis, Operaions Research Cener, MT, Cambridge MA, January [8] C. H. Fine and S. C. Graves, A Tacical Planning Model for manufacuring subcomponens of mainframe compuers, J. Mfg. Oper. Mg., 2, 1989, pp [9] J. S. Hollywood, Performance evaluaion and opimizaion models for processing neworks wih queue-dependen producion quaniies, Ph.D. Thesis, Operaions Research Cener, MT, Cambridge MA, June [10].V. Denardo and Chrisopher. S. Tang, Conrol of a sochasic producion sysem wih esimaed parameers, Managemen Science, 43, 1997, pp [11] U. S. Karmarkar, Capaciy loading and release planning wih workin-progress (WP) and leadimes, J. Mfg. Oper. Mg, 2, 1989, pp [12] S. C. Graves and J. S. Hollywood, A consan-invenory Tacical Planning Model for a job shop, Working paper, January 2001, 30 pp. [13] J.R. Jackson, Neworks of waiing lines, Operaions Research, 5, 1957, pp [14] J.R Jackson, Jobshop-like queuing sysems, Managemen Science, 10, 1967, pp [15] K. D. Gordon, and G. F. Newell. 1967, Closed queuing sysems wih exponenial servers, Operaions Research, 15, 1967, pp [16] J. A. Buzaco and D. D. Yao, Flexible manufacuring sysems: A review of analyical models, Managemen Science, 31, 1985, pp [17] R. Suri and J. L. Sanders, Performance evaluaion of producion neworks, Logisics of Producion and nvenory, S. Graves, A. H. G. Rinnooy Kan and P. Zipkin (eds.), Handbook in Operaions Research and Managemen Science, Vol. 4, 1993, Norh Holland. [18] W. Whi, The Queuing Nework Analyzer, Bell Sys. Tech., 62, 1983, pp [19] W. Whi, Performance of he Queuing Nework Analyzer, Bell Sys. Tech., 62, 1983, pp [20] G. R. Biran and D. Tirupai, Muliproduc queuing neworks wih deerminisic rouings: Decomposiion approach and noion of inference, Managemen Science, 34, 1988, pp [21] J. M. Harrison and R. J. Williams, Brownian moion models of open queuing neworks wih homogeneous cusomer populaions, Sochasics, 22, 1987, pp [22] J. M. Harrison, Brownian models of open queuing neworks wih heerogeneous cusomer populaions, Sochasic Differenial Sysems, Sochasic Conrol Theory and Applicaions, W. Fleming and P. L. Lions (eds), MA Volume 10, Springer-Verlag, New York, 1988, pp [23] L. M. Wein, Dynamic scheduling of a muliclass make-o-sock queue, Operaions Research, 40, 1992, pp [24] A. C. Hax, Aggregae producion planning, Handbook of Operaions Research, Vol. 2, J. Moder and S.. lmaghraby (eds), Von Nosrand, Reinhold. [25] K. Baker, Requiremens Planning, Logisics of Producion and nvenory, S. Graves, A. H. G. Rinnooy Kan and P. Zipkin (eds.), Handbook in Operaions Research and Managemen Science, Vol. 4, 1993, Norh Holland. [26] G. R. Biran and D. Tirupai, Hierarchical Planning, Logisics of Producion and nvenory, S. Graves, A. H. G. Rinnooy Kan and P. Zipkin (eds.), Handbook in Operaions Research and Managemen Science, Vol. 4, 1993, Norh Holland. [27] O. Wigh, npu/oupu Conrol A real handle on lead ime, Producion & nvenory Managemen, 3 rd Qr.