An Extension to the Tactical Planning Model for a Job Shop: Continuous-Tie Control Chee Chong. Teo, Rohit Bhatnagar, and Stephen C. Graves Singapore-MIT Alliance, Nanyang Technological Univ., and Massachusetts Institute Technology Abstract We develop an extension to the tactical planning odel (TPM for a job shop by Graves []. The TPM is a discrete-tie odel in which all transitions occur at the start each tie period. The tie period ust be defined appropriately in order for the odel to be eaningful. Each period ust be short enough so that a job is unlikely to travel through ore than one station in one period. At the sae tie, the tie period needs to be long enough to justify the assuptions continuous workflow and Markovian job oveents. We build an extension to the TPM that overcoes this restriction period sizing by peritting production control over shorter tie intervals. We achieve this by deriving a continuous-tie linear control rule for a single station. We then deterine the first two oents the production level and queue length for the workstation. Index Ters job shop, tactical planning odel, oents production quantities and queue lengths, production soothing I. INTRODUCTION HIS paper considers an extension to the tactical T planning odel a job shop by Graves []. A job shop is a process structure in which there is a wide variety jobs and a jubled work flow through the shop. Due to the large variety jobs and the diverse processing requireents each job, there is no distinct workflow through the shop. Because the wide job variety and thus a lack prevailing work flow, production control is difficult and can be very coplex. A job shop ten represents the ost coplex and generic for a anufacturing environent. Therefore, the ability to plan a job shop will provide useful insights for production control other process structures. Graves [] develops an analytical odel to support tactical planning in job shops. The odel characterizes the Manuscript received 9 th Noveber 24. C.C. Teo is with the Singapore-MIT Alliance (SMA at Nanyang Technological University, 5 Nanyang Avenue, Singapore 639798. (eail: ps75697h@ntu.edu.sg R. Bhatnagar is with the Nanyang Business School, Nanyang Technological University, 5 Nanyang Avenue, Singapore 639798. (eail: ARBHATNAGAR@ntu.edu.sg S. C. Graves is with the Sloan School Manageent and the Engineering Syste Division at MIT, Cabridge MA 239 USA. (eail: sgraves@it.edu The second and third authors are Fellows the Singapore-MIT Alliance. interrelationship variability, production soothness and work-in-process inventory in a job shop. As such, the odel provides a foundation for the understanding for the trade-fs inherent in the planning a job shop. The tactical planning odel (TPM is a discrete-tie odel in which all transitions within the odel are governed by an underlying tie period. The odel assues that all oveent jobs occurs at the start each tie period. As such, one ust set the tie period to be short enough so that it is unlikely for one job to travel through two successive stations in one tie period. The TPM does not explicitly odel the flow discrete jobs, but rather odels the flow work due to the jobs. Work copleted in the current period flows to downstrea stages in the next period. However, in discrete anufacturing, each job is only transferred to the downstrea station upon copletion. So in order to accurately odel the job oveent, the tie period for the TPM should preferably be long to increase the fluidity the flow the discrete jobs. Furtherore, the TPM assues a Markovian workflow. The validity this assuption depends on whether each workstation in the shop produces a stable ix jobs. If any jobs can be copleted in one period, then it is ore likely that there is a stable output; but this also argues for a longer tie period. In this paper we extend the odel in [] to address the odel s liitation due to the setting the period length. In the next section, we give a literature review work related to the TPM. We then present a brief review the TPM in section III. Next, we illustrate in section IV the liitation the TPM due to the restriction sizing the tie periods. In section V, we derive a new linear control rule that will reove this restriction for a single-station syste, and also deterine the first two oents the production quantity and queue length. We conclude in section VI with soe thoughts on future research directions. II. RELATED WORK Graves [] develops the TPM as a tactical planning tool for job shop operations. The TPM is a discrete-tie linearsyste odel that deterines the first two oents the production and queue levels, given the planned lead ties the workstations. The odel tracks the workload at each station rather than the individual jobs; the odel assues
that the volue work arrivals at a station are in fixed proportions the work copleted at upstrea stations. To date, there are several extensions to the TPM. Parrish [2] proposes a fraework for odeling work releases to eet the delivery due date for a finished product. In addition, he also shows how to adjust the control paraeters the TPM to change service easures in eeting deand. Graves [3] presents three extensions to a single-station odel the TPM. First, he odels a station that fails according to a Bernoulli process. Second, he incorporates variability due to lot-sizing, and finally, he presents the atheatical bounds on a station with capacity constraint. Mihara [4] extends the work Graves [3] when he looks at an unreliable ulti-station TPM. Siilar to Graves work, the stations fail according to a Bernoulli process. Fine and Graves [5] test the TPM on a real-life job shop when they apply the TPM to a shop that anufactures theral conduction odules for ainfrae coputers [5]. Here, the odel is extended to allow consideration features such as release policies. The odel is then used to study the ipact various planning policies and the effect changes in product ix. Hollywood [6] deonstrates how to calculate approxiations for the steady-state oents TPM with general non-linear control rules. His odel allows for the odeling achine congestion due to capacity loading. Other efforts adapt the TPM to pull systes. Leong [7] odels a Kanban control syste and other pull systes using the TPM in which work is produced at a station whenever there is a downstrea inventory shortfall. More recently, Graves and Hollywood develop a constantinventory TPM in which the release work into the shop is regulated to aintain a constant inventory level [8]. III. REVIEW OF THE TACTICAL PLANNING MODEL The tactical planning odel (TPM is a discrete-tie, continuous flow odel. All transitions within the odel occur at the start each tie period, and the jobs are odeled as workload easured in tie units (e.g. hours. The workflow is assued to have a Markov property: that is, the processing requireents at a station do not depend on how work got to the station. As such, each individual job has no identity. Central to the TPM odel is the linear control rule, which is stated as Pit = iqit ( where P it is the aount production copleted by work station i in tie period t, Q it is the queue level at the start period t, and the paraeter i, < i, is a soothing paraeter. This rule states that the production P it at workstation i is a fixed portion ( i the queue work Q it at the start the period. In particular, / i represents the nuber periods, on average, the work requires to ove through the work station. We interpret / i to represent the planned lead tie. We can view the control rule in ( as a prescriptive equation, i.e. to preserve the integrity the planned lead tie, we ust shift capacity to heavily loaded stations. But ( can also be considered as a descriptive equation where production resources are naturally flexed to accoodate the varying workloads at the stations. The queue level Q it satisfies the standard inventory balance equation Q it = Qi, t Pi, t + Ait (2 where A it is the aount work that arrives at workstation i at the start period t. By substituting ( into (2, we obtain a first-order soothing equation with i as the soothing paraeter: Pit = ( i Pi, t + i Ait (3 Each workstation can receive two types arrivals; one type arrival consists jobs that have their first processing step at the station, while the other type arrival consists in-process jobs that have just copleted processing at an upstrea station. We odel the arrivals to station i fro another station j by the equation: Aijt = φ ij Pj, t + ε ijt (4 A ijt is the flow work arriving at station i fro station j at the start period t, φ ij is a positive scalar and ε ijt is a rando variable. We assue that one unit (e.g. hour work at station j will trigger, on average, φ ij tie units work at station i. The variable ε ijt is a noise ter that odels uncertainty between production at j and arrivals to i, and is assued to be an i.i.d. rando variable with zero ean and a known variance. The arrival to station i is given by A it = Aijt + Nit (5 j where N it is an i.i.d. rando variable for the workload fro new jobs that enter the shop at station i at tie t. Substituting for A ijt, we obtain Ait = φij Pj, t + ε it, where ε it = Nit + εijt (6 j j The ter ε it represents arrivals that are not predictable fro the production levels the previous period, and consists work fro new jobs and noise in the flow. By assuption, the tie series ε it is independent and identically distributed over tie. We can restate the equations for production (3 and for arriving work (6 in atrix-vector for:
P t = ( I D + DAt, (7 At = Φ P t + ε t (8 where P t = {P t, P nt }', A t = {A t,,a nt }', and ε t = {ε t,,ε nt }' are colun vectors rando variables, n is the nuber workstations, I is the identity atrix, D is a diagonal atrix with {,, n } on the diagonal, and Φ is an n-by-n atrix with eleents φ ij. By substituting equation (8 into equation (7, we find that = ( I D + DΦ + Dε t (9 By iterating this equation and assuing an infinite history the syste, we rewrite P t as an infinite series s = ( I D + DΦ Dε t s ( The ean and the covariance for the noise vector ε t are denoted by µ = {µ,,µ n }', and Σ = {σ ij } respectively. The first two oents P t are given by and where B = I D + DΦ E[ ] = ( I D + DΦ = ( I Φ µ s Dµ ( s s S = var( = B DΣ DB' (2 We note that S provides the variance the production requireents for each station, as well as the covariance for each pair workstations. In addition, we deterine the first two oents the queue levels. Fro (, we note that and Therefore we have Qt = D P t (3 E[ Qt ] = D E[ P t ] (4 Var( Qt = D SD (5 The infinite series in equations (, (2, (4 and (5 converge provided that ρ(φ <, where ρ(φ denotes the spectral radius Φ (see []. IV. LIMITATION OF TPM In this section, we exaine the liitations the TPM due to the sizing the discrete tie period. This provides the otivation for our work, which is presented in the next section. In the TPM, all job oveents can only occur at the start each tie period. In order to odel oveent jobs in the actual shop, we are restricted to set the period length to be short enough so that it is highly iprobable for one job to travel through ore than one station in a single tie period. However, due to the continuous-flow assuption, the tie period should be long relative to the workload the individual discrete jobs. This will increase the fluidity the discrete jobs and will ake the continuous-flow assuption ore reasonable. Now we use an exaple to illustrate the discrepancies between the odel and the actual syste due to the above contradictory objectives in period sizing. To siplify our illustration, we consider a siple syste that consists two stations in series, naely Station i and Station j. We further assue that the planned lead tie Station i is 2 hours, while that Station j is hour. We set the length the tie period to be hour, as we assue that it is unlikely for a job in Station i to travel beyond Station j in hour. Now consider a job that enters the epty syste and arrives at Station i at the start period t. We suppose that the job has a processing tie 2 hours at Station i, and hour at Station j. We illustrate and copare the sequence events for the actual syste and the TPM fro period t to t + 2. Fig. shows the actual syste fro the start period t to t + 2. The job arrives at Station i at the start period t. Since the planned lead tie is 2 periods, the job is processed at Station i till the end t +. The job is then transferred to Station j at the start t + 2. It is then processed at Station j till the end period t + 2 since the planned lead tie is period. Now we look at the sae scenario in the context the TPM. Suppose that both Stations i and j produce according to the TPM control rule in equation (. In this case, given the planned lead tie each station, we have i = ½ and j =. Job oveents between the two stations are odeled by equation (4. We assue that the ter φ ij =.5 given the processing ties the job, and ε ijt =. Fig. 2 illustrates the workflow in the TPM fro the start period t to t + 2. As shown in the figure, Station i processes half the in-queue workload at the start each period ( i = ½, while Station j processes the entire workload ( j =. And the workload processed by Station i in each period generates half the workload at Station j at the start the next period (φ ij =.5.
Station i processes hour work in t Station i processes hour work in t + Station j processes hour work in t + 2 2 hours Station i Station j hour Station i Station j Station i Station j hour t t+ t+2 Fig. Actual syste fro start period t to t + 2.5 hour transferred to Station j at start next period Station i processes Station i processes.5 hour work in t hour work in t +.25 hour transferred to Station j at start next period Station j processes.5 hour work in t + Station i processes.25 hour work in t +2.25 hour transferred to Station j at start next period Station j processes.25 hour work in t + 2 2 hours Station i Station j hour Station i Station j Station i Station j.25 hour.5 hour.5 hour t t+ t+2 Fig. 2. TPM fro start period t to t + 2 processed hour processed.5 hour.25 hour t t + t + 2 tie t t + t + 2 tie Actual syste TPM Fig. 3. Production levels at Station j in actual syste and TPM
In the actual syste, the discrete job oves to the downstrea station only upon copletion. However, in the TPM, work flows as a fluid to the downstrea station even if the discrete job is not copleted. In this exaple, the discrete job is split up and oved in parts to the downstrea station. This is due to the fact that the period length is short ( hour copared to the workload the job at Station i (2 hours. As a result, a workload fro the job is generated at Station j even though it is still inprocess at the upstrea station. Fig. 3 shows the production levels at Station j over period t to t + 2 in both the actual syste and the TPM. The production level in the actual syste consists a spike hour in period t + 2. In the TPM, the production is soothed, with production levels at.5 hour and.25 hour in period t + and t + 2 respectively. This exaple considers a syste only two stations. To odel a coplex job shop using the TPM, one ust set the period length by considering the job oveents between all stations. On the one hand, the period length should be such that it is unlikely that a job copletes processing at ore than one work station in a tie period. In a shop with any stations and where jobs ove quickly between stations, this iplies that the period length be set on the order the average job workload. On the other hand, the accuracy the TPM depends on assuptions continuous workflow. We prefer to set a long tie period relative to the workload the jobs so that the discrete jobs will be ore fluid. Furtherore, the TPM assues a Markovian workflow such that transitions do not depend on the history the syste. In essence, the odel assues that each station processes a relatively stable ix jobs in each tie period, so that subsequent flow to downstrea stations is stable as well. The validity this assuption also depends on the length the tie period. If only a few jobs are copleted in each period, then it is unlikely that there is a very stable output. Therefore, this assuption will be ore valid if we are able to set a longer tie period. In addition, the restriction period sizing ay hinder the application the TPM to production planning and scheduling. In soe job shops, it takes only a short tie (e.g. less than an hour for a job to travel through ore than one station, and thus the discrete tie period has to be short. However, the paraeters in ost planning systes, such as the deand requireents, are defined in daily or weekly tie units. Thus the ability to set a longer tie period will facilitate the application the TPM to production planning and scheduling. In the next section we describe an approach to extend the TPM to be less dependent on the choice tie period. V. MODEL In this section, we develop a single-station odel to overcoe the liitations the TPM discussed in section IV. We will derive a linear control rule that accoodates ore frequent arrivals to the work station; as a consequence, we can now perit work to flow through ore than one work station within a tie period. We find the first two oents the production and queue length variables using the derived control rule. Without loss generality, we suppose that each discrete tie period t has a length one tie unit. We sub-divide each tie period t into equal subintervals sub-period s, where s =, 2,,. We define = /, where is the length each sub-period. We assue that work flow can arrive at the start each sub-period. We also assue that we set the production in each sub-period according to the linear control rule (. Thus, we control the production according to a finer tie grid, and we allow for ore fluid arrivals to the work station. We restate the control rule ( for each sub-period s as Y(, s =.X(, s for s =, 2,, (6 where Y(, s is the production level in sub-period s length, X(, s is the queue length at start sub-period s length and is the soothing paraeter. We interpret / as the planned lead tie; however, we now perit to assue any positive value, and thus, we perit the planned lead tie to be less than one tie period. Equation (6 is analogous to ( such that the production Y(, s in each sub-period s is a fixed fraction the queue length X(, s at the start each sub-period. Now we proceed to develop the linear control rule for P t in ters Q t and A t. These variables have the sae definition as in TPM: P t is the production copleted in period t, Q t is the queue length at the start period t, and A t is the arrival work to the station in period t. However, we now assue that A t does not arrive at the start the period, but rather arrives uniforly over period t. In particular, we assue that in each sub-period, the arrival aount is equal to A t /. We have the following boundary condition for the queue length for the first sub-period: X(, s = = Q t + A t / For s >, we odel the queue length in the sub-period s by the standard inventory equation X(, s = X(, s - - Y(, s - + A t / (7 By substituting (6 into (7, we obtain X(, s = ( - X(, s - + A t / (8 Now to get an expression for P t, we note that = Y (, s = s (9
We su the above expressions for X(, s to find X (, s + ( s (2 Fro (2, we observe that s + ( s ( s ( (2 We now cobine (9 and (2 to get ( (22 Fro (22, in order to get an expression for P t, we need to find X(,. Fro (8 and repeated substitution, we can then write = ( Qt + At ( + ( +... + ( We can use (23 to re-write (22 as = ( ( Qt. ( ( At + Thus we can write (24 as where and (23 (24 P t = β ( Qt + γ ( At (25 β ( = ( γ ( = = β ( ( ( Now we proceed to deterine the continuous-tie liits for β( and γ( as the length the sub-period goes to zero. This corresponds to a continuous-tie control in which the production level will satisfy (6 at every instant in tie. We use the forula li ( x x = e to x obtain the continuous-tie liit β( : β β = li ( = li [ ( = li [ ( ] = e For γ(, we find that γ = li γ ( = li = li = li = ] ( ( ( ( + li [ ( ( ] ( ( β ( e = We can now restate (25 for the continuous-tie control as: = β Qt + γat (26 where β and γ are given above. The balance equation for the queue length for the single station is now given by: Q t. (27 This balance equation differs fro (2 in the TPM, due to the new assuption that arrivals occur continuously throughout a period. Hence, we define Q t to be the queue length at the start period t, prior to any arrivals in period t. By substituting (26 into (27 and repeated substitution, we obtain: i Qt = ( γ ( β At i. (28 i= If we assue that the arrivals are i.i.d. with ean µ and variance σ 2, then we find the two oents for the queue length fro (28: E [ Qt γ ] = µ = β µ 2 2 ( γ σ Var ( Q t = 2 2β β
Siilarly, we obtain the two oents for the production variable: E [ P t ] = µ β 2 2 2 Var ( P t = ( γ + γ σ 2 β Thus, we have analytical expressions for the oents these two variables. We have expressed these in ters the paraeters β and γ, both which are functions our soothing paraeter. However, it is not obvious how the variances production and the queue length depend on the soothing paraeter. To provide soe insight into this, we graph each variance as a function the soothing paraeter in Fig. 4 (with σ =. We see fro these graphs that the variance production drops as we do ore soothing, i.e., sall values, or equivalently longer planned lead ties. However, as we do ore soothing production, both the expected queue length and its variance grow..8.6.4.2 Var (P(t Var (Q(t.5.5 2 2.5 3 Fig. 4. Variances P t and Q t as a function for σ = These oents provide a siple way to see the fundaental trade-fs across the three eleents tie, capacity and variability. For given level deand variability (given by the arrival process, as we reduce the planned lead tie, production becoes ore variable and ore capacity is required; alternatively, as we sooth production, we need less capacity but ore tie in ters the planned lead tie. These insights are the sae as for the TPM. But the odel given here is ore general in that we perit continuous arrivals to the work station and continuous production control. VI. CONCLUDING REMARKS In this paper, we extend the TPM to address its dependency on the choice tie period; in particular, we perit work to arrive throughout a period, as opposed to at the start each period and we perit continuous production control. We derive a linear control rule by assuing that the production is controlled according to a finer tie grid. We then obtain the continuous-tie liits the control rule, and find the first two oents the production and queue rando variables based on the control rule. Our next step is to extend the single-station odel to a network workstations as would exist in a job shop. We will need to deterine how best to odel the flow work between stations, so as to capture actual behavior and yet still retain analytical tractability. We will need to investigate the stability this odel; in particular, we wish to find the conditions convergence for the first two oents production and queue vectors. In the TPM in [], the first two oents converge provided that the spectral radius workflow atrix Φ is less than. This iplies that a unit work at any station cannot eventually result in ore than one unit work for the sae station. It is interest to evaluate the convergence conditions this odel as copared to the TPM in []. In addition, we have not established the benefits this extension over the original TPM under different job shop conditions. It is worthwhile to carry out a study, perhaps through a coputational experient, to copare how this extension perfors relative to the TPM in different operating conditions, such as the stability job ix and speed job oveent between stations. Another opportunity for future research is to extend this odel to incorporate congestion effects due to capacity loading. This would involve relaxing the linear control assuptions for each station. Hollywood [6] has developed an approxiation for the TPM with nonlinear control rules. It ay be possible to integrate his approxiation into this odel. REFERENCES [] S.C. Graves, A Tactical Planning Model for a job shop, Operations Research, 34, 986, pp 522-533. [2] S.H. Parrish, Extensions to a odel for tactical planning in a job shop Environent, S.M. Thesis, Operations Research Center, Massachusetts Institute Technology, June 987. [3] S.C. Graves, Extensions to a Tactical Planning Model for a job shop, Proceedings the 27 th IEEE Conference on Decision and Control, Austin, Texas, Deceber 988. [4] S. Mihara, A Tactical Planning Model for a job shop with unreliable work stations and capacity constraints, S.M. Thesis, Operations Research Center, MIT, Cabridge MA, January 988. [5] C. H. Fine and S. C. Graves, A Tactical Planning Model for anufacturing subcoponents ainfrae coputers, J. Mfg. Oper. Mgt., 2, 989, pp 4-34. [6] Hollywood, John S. 2. An Approxiate Planning Model for a Job Shop with Nonlinear Production Rules, working paper, 32 pp. [7] T.Y. Leong, A Tactical Planning Model for a ixed push and pull syste, Ph.D. progra second year paper, Sloan School Manageent, Massachusetts Institute Technology, July 987. [8] S. C. Graves and J. S. Hollywood, A constant-inventory Tactical Planning Model for a job shop, Working paper, January 2, revised March 24, 35 pp.