Order Sequencing and Capacity Balancing in Synchronous Manufacturing

Transcription

1 Order Sequencing and Capacity Balancing in Synchronous Manufacturing Jan Riezebos To cite this version: Jan Riezebos. Order Sequencing and Capacity Balancing in Synchronous Manufacturing. International Journal of Production Research, Taylor & Francis, 00, pp.. <0.00/00000>. <hal-00> HAL Id: hal-00 Subitted on 0 Feb 0 HAL is a ulti-disciplinary open access archive for the deposit and disseination of scientific research docuents, whether they are published or not. The docuents ay coe fro teaching and research institutions in France or abroad, or fro public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de docuents scientifiques de niveau recherche, publiés ou non, éanant des établisseents d enseigneent et de recherche français ou étrangers, des laboratoires publics ou privés.

2 International Journal of Production Research Order Sequencing and Capacity Balancing in Synchronous Manufacturing Journal: International Journal of Production Research Manuscript ID: TPRS-00-IJPR-0.R Manuscript Type: Original Manuscript Date Subitted by the Author: 0-May-00 Coplete List of Authors: Riezebos, Jan; University of Groningen, Faculty of Econoics and Business Keywords: SYNCHRONOUS MANUFACTURING, SEQUENCING, ORDER RELEASE, HEURISTICS, INTEGER PROGRAMMING Keywords (user): SYNCHRONOUS MANUFACTURING, SEQUENCING

3 Page of International Journal of Production Research Order Sequencing and Capacity Balancing in Synchronous Manufacturing JAN RIEZEBOS * Departent of Operations, Faculty of Econoics and Business, University of Groningen, The Netherlands * Eail: j.riezebos@rug.nl

4 International Journal of Production Research Page of Order Sequencing and Capacity Balancing in Synchronous Manufacturing Abstract Synchronous anufacturing ais at achieving the benefits of interittent production lines in production situations that operate without lines. Benefits such as short and constant throughput ties and predictable capacity loading can be acquired through an appropriate design of the synchronous anufacturing syste and its control syste. The order release echanis is an essential part of this control syste. It deterines the sequence in which orders are released to the shop floor. As orders ay differ in the aount and distribution of their capacity requireents over subsequent production stages, total capacity load ay vary over tie. If the available capacity per period is not flexible, capacity balancing becoes an issue in the order release decision. In practice, heuristics or rules of thub are used to solve this proble, but their effectiveness is questioned. This paper exaines the effectiveness of soe new heuristics that are based on insights fro assebly syste design and work load control, and copare their perforance with an optial solution approach. The approaches are evaluated in a rolling schedule environent, and under different levels of capacity fluctuations and proble sizes. The results show that the perforance of the heuristic solutions deteriorates if capacity fluctuations between the stages increase. If we easure both the aount and frequency of shortages over a long period of tie in a rolling schedule environent, a quite siple rule that only takes the available capacity during the first stage into account outperfors ore intelligent rules. Keywords: Synchronous anufacturing; order release; sequencing, heuristics; integer prograing Introduction Synchronous anufacturing is a concept that is behind several popular shop floor anageent approaches, such as Lean and Quick Response Manufacturing. The basic idea is to create a flow of work through the anufacturing syste, either continuous or interittent, in order to achieve short and constant throughput ties and a predictable loading of the resources in the syste. Both the structure of the production syste and the control syste are to be designed such that this flow can be achieved (Schenner and Swink ). One of the earliest and ost faous synchronization ethods is single product assebly line balancing (see Scholl and Becker (00) for an overview). The production syste is designed as a connected line, progress control is eliinated at all, and the order release decision reduces to the decision on the cycle tie of the line, i.e. the tie between subsequent releases. The concept was applied first in the high volue autoobile plant of Henry Ford. As changing the cycle tie (a control decision) ay require a different design of the production line (i.e., nuber of stations and assignent of tasks), the relation between both design decisions is central in the assebly line balancing proble. More recent approaches give attention to the issues in case the anufacturing process is not connected as a single product line. For ulti-product lines, the concept of assebly line balancing has been extended to ixed and ulti odel assebly line balancing and generalized (Becker and Scholl 00). If the product(s) are no longer produced within a single line but in several stages of production, a control syste can be used to connect several autonoous parts of the anufacturing syste. Period Batch Control (PBC) (Burbidge,, Benders and Riezebos 00) is to be considered as the earliest exaple in this category. PBC

5 Page of International Journal of Production Research synchronizes the assebly stage with one or ore preceding and feeding stages of production, such as parts anufacturing and aterial acquisition. Ahadi and Wurgraft () give attention to the issue of designing such anufacturing systes. The control syste becoes ore essential in achieving a synchronized flow of work through the anufacturing syste. Control within a stage of production is often decentralized to relatively autonoous production units, denoted as cells or teas. Here the concept of cellular anufacturing or tea oriented assebly eerges (Bukchin and Masin 00). PBC provides the control between the stages through short but fixed offset ties that are identical for all stages. Orders are released at fixed tie intervals. The resulting capacity load variation over tie is predictable and anaged through the order release decision and capacity anageent instruents, as sufficient tie reains for capacity level changes and outsourcing decisions. An iportant characteristic of this synchronization approach is the use of a fixed cycle tie for each stage. Other approaches, such as the Kanban syste and the Dru Buffer Rope syste, do not use a fixed cycle tie for each stage, which coplicates the capacity anageent proble. They focus on the order release decision and use a type of aster production schedule (denoted as a level schedule in Kanban systes (Miltenburg ), and as a dru schedule in DBR systes (Riezebos et al. 00)) to balance the load for specific stages over tie. Recently, fixed cycle-tie synchronization approaches have been developed that no longer iplicitly assue an inflexible capacity. They consider total capacity to be liited, but capacity to be flexible between the stages of the production syste. The reason for relaxing this assuption is that workers are nowadays increasingly ulti-skilled and cross-trained. This flexibility akes it possible to handle capacity fluctuations between stages in a production syste. However, total capacity in ters of the nuber of workers available does not increase through such easures. Therefore, these synchronization approaches ai at an order release decision that effectively uses the available capacity of the ulti-skilled workers while still realizing a high output for the whole production syste. Exaples of papers in this area are Lee and Vairaktarakis (), Vairaktarakis et al. (00), Gronalt and Hartl (00), Bukchin and Masin (00), and Cevikcan et al. (00). The coplexity of the resulting synchronization probles has been analyzed by Vairaktarakis and Cai (00). They showed that ost levelling probles in such systes are NP-coplete, even if they only consist of two stages. Therefore, in practice heuristic solutions are being used to solve the order release decision, as the nuber of stages often is uch larger than two. This paper discusses various single pass heuristics for the order release decision in such a synchronization approach with a fixed cycle tie in a ulti-product ulti-stage situation. Single pass heuristics deterine a sequence without back tracking or pair wise interchanging parts of a solution. Such heuristics are often used in practice. The question is whether these heuristics can be iproved by incorporating insights fro related fields, such as workload control and assebly line balancing. Moreover, the perforance of these heuristics ay or ay not be sensitive to proble size and capacity fluctuation. This paper will exaine the perforance of several heuristics in a rolling schedule environent as faced by firs that apply such a synchronization approach. Testing in a rolling schedule environent provides better insights in the perforance of these heuristics in the long ter, as it prohibits the negative ipact of postponing probles to the end of the cycle. This paper is organized as follows. We introduce the proble by discussing an illustrative proble, based on the case of a jewellery fir in Ssection. Section presents the heuristics that we developed for supporting the order release decision. Section develops a atheatical prograing odel that deterines the sequence of order releases. This odel not only ais at a feasible solution for the near future, but takes also in account the possibility of achieving feasible solutions behind the planning horizon. Section evaluates the perforance of both approaches using a siulation approach. It describes the experiental design and presents the results. Section gives our conclusions and suggestions for future research.

6 International Journal of Production Research Page of Proble definition An illustration of the order sequencing and capacity balancing proble is found in the longitudinal study of Süer (see e.g. Süer and Gonzalez, Süer ) on a jewellery fir. Production of jewellery requires siilar operations for various product types, such as earrings, rings, bracelets and chains. The processing plans ay vary per product type in ters of type of plating. Most operations are perfored anually, using siple tools, sall achines and equipent. Exaples are casting, cleaning, and tubling. The eployees of a cell have been trained to perfor all tasks in all stages of the production process. Speed differences between eployees are inor. Capacity anageent is therefore ainly oriented towards work force scheduling, and hiring/firing decisions. The fir operates in synchronous anufacturing ode. The stages represent a subset of operations that are to be perfored within a single period of tie (four hours in case of the jewellery fir). At the end of each period, the various subsets of operations all need to be copleted, as all jobs that are in progress are transferred to their next stage at the sae oent in tie. At such a transfer oent, eployees ay have to switch to other tasks within the cell. Soe operations, such as tubling and plating, are necessarily perfored in batch. The inial throughput tie for a batch of products deterines the length of the period that can be used in a synchronous anufacturing syste. The cell starts each period with a new order that is selected fro a list of orders that should be copleted during the cycle (i.e., at the end of the week). The batch size, process plans, and work content per stage ay differ per order. Therefore, capacity requireents ay vary strongly, both per order and per stage. The expected throughput tie of a batch within a stage is a function of the nuber of eployees allocated to this stage. In a synchronous anufacturing syste, the inial nuber of eployees required in each stage in order to finish this batch within the end of each period can be calculated. An iportant proble faced in such synchronous anufacturing systes concerns the capacity balance over tie (Cesani and Steudel 00). A cellular syste akes it less appropriate to vary the total nuber of eployees over tie. Eployees should feel theselves responsible for the whole task of the cell. A relatively constant nuber of eployees over various periods is therefore preferred. There are two planning decisions that affect the objective of achieving a unifor loading of total capacity in ulti-odel systes. The batch-size decision, and the order sequencing or release decision (Karabati and Sayin 00). This paper will give attention to the latter decision. The release decision can be supported either by intelligent design of an order sequencing heuristic, or by creating beforehand one or ore feasible order release sequences fro which the cell can choose. This paper copares these two approaches for direct support of the order release decision.. Illustrative exaple Table presents a realistic exaple with 0 orders that have to be released during one week. We have 0 orders (A,,J) and five stages j=,,. An order that starts in period t= in stage j= arrives in period t= in stage j=, and leaves the syste at the end of period t=, if stage j= has been finished. Orders are represented using different shades. The aount of capacity required in a stage differs per order and is presented in the cells of the table. A row shows the fluctuating capacity requireents of that stage. Orders ay also require a different total nuber of eployees (i.e. the su of the cells with identical shade). However, the su of the cells in the sae colun is ore iportant for the capacity anageent of the fir. This shows the total nuber of eployees needed in a single period. If this nuber exceeds the available capacity, the fir has to hire additional eployees or change the sequence of orders. The proble is to deterine one or ore sequences for the orders A,,J such that the available capacity of 0 eployees in each period t is not exceeded as long as possible.

7 Page of International Journal of Production Research For the first four periods, earlier decisions on the sequence in a previous week affect the loading of the available capacity. The cells at the botto-left side express the capacity requireents of these already started orders that still have to coplete one or ore stages. The sae effect appears at the end of the week, as the last four orders in the sequence affect not only capacity in this week, but also capacity requireents in the next week. Here the end-of-horizon effect or truncated-horizon effect appears (see e.g. Baker, Stadler 000). ***INSERT TABLE ABOUT HERE*** Literature uses various approaches to cope with the end-of-horizon effect. Soe papers (e.g., Ding and Cheng, Lee and Vairaktarakis, Ouenniche and Boctor ) totally neglect the end-of-horizon effect. Others (e.g., Karabati and Sayin 00, Vairaktarakis and Cai 00) assue that exactly the sae order release pattern will be repeated during the next week. These ethods do not address the end-of-horizon effect in a realistic way. Gronalt and Hartl (00) introduce the overall average worker assignent in their nuerical experients. We propose a slightly different approach that uses the stage-average capacity requireents as capacity load factor if the actual capacity load is unknown. According to Stadler (000), this provides a terinal condition that iproves the quality of the solutions in a rolling schedule environent. The upperright side of Table shows the average required capacity per stage. For exaple, stage j= has an average =. eployees, which is used to estiate capacity load of ( ) the load in stage j= in periods t=-. Note that in Table no order exceeds the capacity liit of 0 eployees. For exaple, order A requires ++++=0, and J requires ++++= eployees. The ean required capacity per order equals. eployees. The loading should result in a sequence for which the available capacity per period is not exceeded. The sequence presented in Table results in four periods that encounter a capacity shortage: t=,,,0. Can a better order sequence be found?. Matheatical proble forulation Let i=,,n index for orders j=,, index for stages t=,,n+- index for periods Paraeters C ij Capacity requireents to coplete stage j of order i i=,,n; j=,, CR Capacity requireents to coplete stage j in period t for an order fro a forer cycle tj t=,,-; j=t+,, ARC j Average required capacity in stage j = AC t j CR + t= tj i= n+ j j=,,- Available capacity in period t (assuing the capacity is fully cross-trained) t=,,n+- n C ij

8 International Journal of Production Research Page of w t Weight factor for capacity shortage in period t (i.e. wt = if t=,..., n; wt if t> n ) t=,,n+- Decision variable X it = if order i starts in period t (deterines the sequence) =0 else i=,,n; t=,,n Interediate variables Y tj Capacity requireents to coplete stage j in period t t=,,n+-; j=,, CS t Capacity shortage (expected) in period t = ax( 0, j= Ytj ACt) t=,,n+- The proble can now be forulated as: Given: a set of i=,,n orders that have to start during periods t=,,n; their capacity requireents Cij during j=,, stages of copletion; the capacity requireents CRtj for orders that have already started but are not yet copleted; available capacity per period AC t, deterine X ( i,..., n; t,..., n) such that it n+ t= = = (the sequence of the n orders) w CS t t is iniized. **INSERT FIGURE ABOUT HERE*** Figure provides an overview of the variables and paraeters in ters of the structure of Table. Note that the value of the variable Y tj in this table is area-specific. In the lower-left side of the table, Ytj is deterined by the given capacity requireents CRtj of already started but not yet copleted orders. In the upper-right side of the table, it is estiated by calculating the average required capacity in the stage. In the reaining part of the table, it is calculated as a linear function of the order sequence decision and the known capacity requireent of the orders. We use the following expression for calculating this stage load: n Yt+ j, j = X,...,,..., i it Cij t= n j= () = Analogues to job scheduling, this proble can be viewed as a weighted tardiness proble, where tardiness is not related to exceeding the due date of a job but to exceeding the available capacity in a period. Morton and Pentico (: p. ) denote that probles using this objective are very difficult to solve exactly. The

9 Page of International Journal of Production Research coplexity of this proble has not yet been deterined, but siilar probles that are discussed in Vairaktarakis and Cai (00) are NP coplete in the strong sense for situations with the nuber of stages greater or equal to three. Order release heuristics This section describes several single pass heuristics to deterine a sequence of orders for release. Single pass heuristics do not reconsider decisions taken during the procedure. The heuristics are based on insights fro assebly syste design and work load control. We will introduce the heuristics and show the results of applying the heuristic on the data of Table. FillCap This heuristic is actually being applied in the jewellery fir. It chooses at the start of period t fro the set of assignable orders the order that axially fills capacity in that period. The set of assignable orders contains all orders that still have to be released and have a capacity requireent in stage j= nearest to but not exceeding the available capacity. If there are ore orders to select fro, it chooses randoly. We denote this heuristic as FillCap. See the Appendix for a pseudo-coded description of the heuristic. Fro Table we see that the heuristic FillCap does not take into account the effect of a selected order on the available capacity for the next - periods. As a consequence, in period a capacity shortage occurs. *** INSERT TABLE ABOUT HERE *** AvgLoad The AvgLoad heuristic applies the principle of bottleneck loading that is known fro ixed odel assebly line balancing (e.g. Askin and Standridge : -). The idea is to select the job that will result in a future load on the bottleneck as close to its average rate as possible. Each new assignent will try to correct the gap between the average rate and the cuulative load on the bottleneck that has resulted fro earlier assignents. The bottleneck of this synchronous anufacturing syste is not a single stage, as is usually the case in ixed odel line assebly probles, but the nuber of eployees available. The heuristic should select an order that (if possible) can be assigned to the first period and that brings the total released workload as close as possible to the long tie average. The calculation of the long tie average uses all inforation available, i.e. the load of the set orders to be released in this cycle (each order requires on average. eployee periods) and the load of the already released orders fro the forer cycle (these four orders generate a total workload of eployee periods during 0 buckets (+++). As each order generates workload during buckets (= stages), the backlog fro the preceding cycle corresponds to coplete orders. The average load of an order on the bottleneck equals (0*. + ) / (0+) =. eployees per order. So for the first release decision, we would like to choose an order with a total workload as near as possible to the average rate inus already released load, i.e. *. =.. After selecting the order (either D or I), the load aied at for the next release decision is calculated. We denote this heuristic as AvgLoad. See Table for the results in the exaple proble. StageLoad The heuristic StageLoad ais at achieving a balanced workload distribution over the stages. It is based on insights fro the field of workload control (Land and Gaalan, Riezebos et al. 00, Stevenson et al. 00). It searches for an order that fits best within the expected peaks and troughs of the already generated workload distribution fro the past assignents. The order to be selected should have a dissiilar workload pattern as copared to the already generated pattern. This generates a less variable

10 International Journal of Production Research Page of stageload pattern, which also reduces the variability of the su of the stageloads. In this way we ight be able to avoid suddenly appearing large fluctuations in the required nuber of eployees. StageLoad considers the release of one order as a release of separate workloads for the stages. Each one of the should receive a workload that brings the total as close to the desired average aount as possible. StageLoad selects the order that iniizes the su of these quadratic distances. The su of squares in the first step of StageLoad is inial for order C, so C is chosen. See Table for the results on the exaple proble. By choosing the order with inial quadratic su of distances to the average stage loads, the creation of sudden peaks and troughs in capacity load is avoided. However, this does not guarantee that sufficient capacity will always be available. For exaple, in period order B is selected, as this order realizes the best stage load distribution, but this order creates a conflict with the available capacity in period. AvailStageLoad The fourth heuristic tries to avoid the conflicts with available capacity and applies the StageLoad heuristic only to the set of orders that fit within the available capacity. Note that the FillCap and AvgLoad heuristic do the sae. If no orders fit within the available capacity, the whole set of unassigned orders is used to deterine the order that fits best. The heuristic AvailStageLoad therefore cobines the two approaches. The results of the heuristic AvailStageLoad in the exaple proble are exactly equal to heuristic StageLoad, as in each stage the preferred order belongs to the set of orders fro which it should be chosen (see Table ). In general, the perforance of heuristic AvailStageLoad is expected to be better than heuristic StageLoad. The coputational coplexity of all four heuristics is O( n ), as in a single pass heuristic the whole cycle is repeated n- ties, and step - is of order n. The first heuristic FillCap is a siple straightforward rule. The other three are ore sophisticated in their coputations, as they ai at achieving not only a short-ter advantage, but a long-ter balance as well. However, they allow a lower than possible fill in the current period in order to achieve the long-ter balance. No guarantee is given that any of the heuristics is able to find an optial solution. For the exaple proble, all four heuristics were unable to find an order sequence without violating the capacity constraint in at least one period. This raises the question on the possibility of finding a sequence that results in a feasible solution. The total nuber of sequences is n!. Coplete enueration is therefore no realistic alternative for large n. However, the advantages of a good order sequence for the productivity and acceptance of this synchronous anufacturing syste akes it worthwhile to search for an iproved solution procedure. The next section will proceed with such a procedure. A atheatical odel to support the order release decision This section discusses a atheatical odel that generates an optial order sequence for all orders that have to be released in the cycle. The atheatical odel is as follows:

11 Page of International Journal of Production Research Miniize suchthat : n i= n t = n+ t= t t X = t=,..., n (each period exactly one order has to start) () it X = i=,..., n (each order is assigned to one starting period) () it Y = CR j=,..., t=,..., j (stage load already deterined in forer cycle) () tj tj w CS Y = ARC t= n+,..., n+ (stage load in future cycles assued tj t+ j, j it ij i= j n j=,..., t n to be equal to the average load) () Y = X C t=,..., n j=,..., (stage load resulting fro release sequence) () j= Y CS AC t=,..., n+ (exceeding available capacity results in shortage) () tj t t [ ] X 0, ; CS 0; Y 0; () it t tj This atheatical odel uses X it as binary decision variable, indicating that order i starts in period t whenever the variable has value. The other variables are non-negative and continuous (see ()). Constraints () and () guarantee that a feasible assignent is obtained. The third constraint () shows that we use this odel in a rolling schedule environent, as we take the effects of the release decisions in the forer cycle into account in the current decision. Constraint () calculates the consequences for future periods that will be affected by the current decision. We use the average load in a stage if the actual load is not being deterined in the current decision round. Constraint () deterines the consequences of the release decision for the stage load in subsequent periods and is siilar to Forula (). Constraint () calculates the capacity shortage per period. The objective function () iniizes a weighted su of the capacity shortages over tie. Through the use of positive weights < for periods behind the sequence horizon, the end of horizon effect (Stadtler 000) is avoided. Most sequencing algoriths that are known fro assebly line balancing (Scholl and Becker 00) or suggested for synchronous systes (e.g. Vairaktarakis et al. 00, Gronalt and Hartl 00) do not treat the end-of-horizon proble. As a result, infeasible solutions ay be encountered at the next decision oent. Soeties the special case of a cyclic schedule (Vairaktarakis and Cai 00) is used to tackle this proble. Our atheatical odel avoids this effect by taking into consideration that the loading in the periods behind the horizon should not exceed the on-average available reaining capacity. Note that the proposed atheatical odel is rather easy to read due to the introduction of the interediate variable Y tj. This holds especially true for odelling the stage load resulting fro the release sequence (). The odel has been ipleented in Lingo in order to find a solution for the proble of Table. As weights we selected wt = for t n; wt = 0. for t> n. The last colun of Table shows the results. During the first ten periods no shortages occur, neither are they expected for the periods -, i.e. for the periods behind the sequence horizon. Hence we conclude that none of the four heuristics achieved an optial result. The question is whether there are differences in the perforance between these heuristics. ()

12 International Journal of Production Research Page 0 of Experiental design and siulation results. Experiental design The forer sections have introduced the four heuristics and the atheatical odel and illustrated their perforance using a realistic exaple. We would like to extend this coparison to a rolling schedule context, as this is the usual situation where such a solution procedure is being applied in synchronous anufacturing systes. The basic preise we would like to test is whether the solution approaches perfor different when the distribution of workload over the stages changes ( MixVar f ( σ Cij ) = ) or the variation in the total workload per order increases ( VolVar= f σ j= C ). The first is a ix-oriented factor, while the second is a volueoriented factor. In addition, we would like to test whether the perforance of the approaches under these ij experiental settings changes if production characteristics (the nuber of stages ) or planning characteristics (the schedule horizon n) are odified. Table shows the full factorial experiental design. *** INSERT TABLE ABOUT HERE*** Each of the configurations (experients) is replicated 00 ties with a run length (rolling schedule) of 0 cycles. For each cycle a rando proble is generated with n orders and stages. The total workload per order and the distribution of this workload over the stages varies randoly according to the specified factorial design of the experient. The proble that has to be solved consists of deterining a sequence for the orders. The consequence of the chosen sequence is input to the scheduling proble of the sae solution approach for the next cycle, as we test the approaches in a rolling scheduling environent. So, although the approaches receive the sae set of new orders at the start of a new cycle, only the first sequencing proble in a replication is copletely identical for all solution approaches. Since we do not know if the size of the proble atters, we have tested the approaches using low and high values for both the scheduling horizon/nuber of orders n and the nuber of stages. In total **00*0 = 0000 sets of orders were generated and the five solution approaches solved the resulting probles. We evaluate the approaches first on the average total capacity shortage per cycle. This is a kind of tardiness easure, as capacity shortage is zero if there is overcapacity in every bucket. For each bucket t=,,n we calculate the capacity shortage CS t and add it to the total capacity shortage in that cycle. The average total capacity shortage is calculated over the nuber of cycles in the rolling schedule. Secondly, the frequency of a shortage in a schedule of one cycle is easured. If in any of the buckets a shortage occurs, this cycle is counted as one with shortages. The ean frequency per replication of 0 runs was calculated. This perforance indicator shows how often a schedule that is constructed with this approach will face a shortage. The ratio of the two perforance indicators (Exp.Shortage = Shortage / Frequency) reveals the ean total aount of capacity short in a cycle that faces a shortage. Available capacity is constant at 0 units per period. Average capacity requireent per order is also constant and set at for all configurations. At a low volue variation, the actual capacity requireent is ±, while a high volue variation has actual requireents equal to ±. The capacity requireent of an order is therefore either in the interval [,] or [,], depending on the value of VolVar. We use a unifor distribution within an interval, so each eleent in the set has equal probability. As the axial ix variation

13 Page of International Journal of Production Research in a stage, MV, ay change in the course of the procedure, we first select randoly a stage and next deterine the nuber of units in that stage. This nuber has to be at least one, as described in the sapling procedure (see Appendix). The proble generator and the five solution approaches have been prograed in Delphi, where the procedure that deterines the optial solution to the ixed integer prograing proble uses the dll of the Lingo solver. Generating these 0000 probles and solving the with the five solution approaches took. hours CPU tie on a. GHz Pentiu IV PC running Windows XP, the ajority of it being used by the optiization approach. The average CPU tie of solving a proble to optiality was. seconds. The n+ = took on average 0. seconds CPU tie. The use of this version of Lingo in the sallest probles ( ) rolling schedule experiental design did not allow us to solve probles with n+ >.. Expectations with respect to the results The solution approaches are expected to perfor differently according to the variation in the experiental design factors. We nuber our expectations in order to enable testing in the next Section. (i) The heuristic FillCap is expected to perfor better if MixVar is high instead of low. (ii) The effect of an increased variation in the total workload (VolVar high) on the perforance of FillCap is uncertain. The reason for (i) is that a high MixVar will result in a higher chance of orders available that fit within the available capacity, even if the aount of capacity available is sall. (iii) Copared to FillCap, AvgLoad is on average expected to result in a lower shortage, while frequency is not changed (iv) AvgLoad will perfor better if MixVar is low instead of high. (v) AvgLoad will be ore effective if VolVar is high. The heuristic AvgLoad iproves the selection procedure of FillCap, but allows idle tie earlier in the cycle as it favours a total workload near the average load on the bottleneck. The effect of an increased variation in the total workload (VolVar high) on the perforance of FillCap is uncertain. The reason for (iv) is that AvgLoad does not take the distribution of workload over the stages into account. A higher variation in this distribution will result in relatively ore shortages. Further, as it ais at balancing the total volue, it will be ore effective if VolVar is high (v). (vi) StageLoad is not effective in anaging shortage and frequency. (vii) StageLoad is ore effective for high MixVar. We do not expect the heuristic StageLoad to be effective in anaging shortage and frequency (vi), as it allows choosing an order while knowing that a shortage occurs. However, it ight result in a ore stable loading of the stages, which is ore effective for high MixVar (vii). (viii) AvailStageLoad is expected to perfor better than StageLoad (ix) AvailStageload is ore effective if MixVar is high. The heuristic AvailStageLoad uses the basic idea of StageLoad in selecting an appropriate order that fits in the available capacity. It is expected to perfor better than StageLoad (viii) and iprove if MixVar is high (ix). (x) Zero shortage solutions are less easy to find in case either MixVar or VolVar is high. In case either MixVar or VolVar is high, we expect that it will be ore difficult to find a solution with zero shortage (x).

14 International Journal of Production Research Page of (xi) Both the average shortage per cycle as the frequency of shortages will increase if n increases. (xii) The average shortage per bucket decreases if n increases. (xiii) Only StageLoad and AvailStageLoad benefit fro an increase in. The experiental variable n is expected to have an equal effect on all approaches. If the length of the horizon n increases, both the average shortage per cycle and the frequency of shortages are expected to increase (xi). However, longer cycles (n ) eans a larger set of orders to select for release, and hence ore possibilities to reduce or avoid shortages. So the increase ight not be proportional to the increase in n, in other words the average shortage per bucket ight decrease (xii). More stages ( ) increases the sensitivity for unbalances between the stages. We therefore expect a perforance iproveent only for the heuristics that take this into account (i.e., StageLoad and AvailStageLoad) (xiii). Next subsection will describe the results of the experients, which enables us to deterine if these expectations are valid.. Results The results of the experients were analyzed on differences in the ean perforance. We used ANOVA to test whether the eans were equal or not. Table shows significant differences with respect to the ean shortage for all ain effects and all interaction effects except one. Most contributing to differences in the ean are MixVar, Approach, and the cobined effect of these two factors. For the frequency of shortages, the sae factors are ost iportant in explaining the total variation. Note that the size of the proble (n and ) has a significant, but inor contribution to explaining the variance in the data. We conclude that the ean perforance of the various approaches cannot be assued to be equal for all factorial levels. ***INSERT TABLE ABOUT HERE*** In order to see if the perforance of all approaches differs significantly (expectations iii, vi, and viii), we applied ultiple coparisons and tested for significant differences using the Tahane test statistic. Table shows the results of this coparison. The type of approach used shows up to be a highly significant factor, as the difference with all other approaches is significant. We only present the differences between Optial and the other approaches, but the sae holds true for the reaining coparisons. Note that Table presents the average differences, based on all values of the experiental variables n,, VolVar, and MixVar. ***INSERT TABLE ABOUT HERE*** The optial solution clearly outperfors the other heuristics. The difference in ean shortage with FillCap equals. eployee periods per cycle, while the frequency of cycles with a shortage is.% higher, as can be seen fro Table (colun Mean Difference (I-J)). AvgLoad perfors even worse -contrary to what we expected in (iii), but still better than the AvailStageLoad heuristic. The StageLoad heuristic perfors worst, as was expected in (vi) and (viii)). However, the good perforance results of FillCap were not expected at all. For all factorial levels, the less sophisticated FillCap appears to be significantly better than the other heuristics that ai at an iproved selection process. FillCap just chooses the order that fills the available capacity in the first stage, and this results both in a lower average capacity shortage and less frequent shortages than ore sophisticated heuristic selection procedures. Sophistication in these single pass heuristics does not see to be of value. ***INSERT TABLE ABOUT HERE*** The question now reains how the solution approaches perfor at the two levels of the experiental factors. The descriptive statistics in Table show that an increase in MixVar has a uch stronger ipact on the extra capacity required and the frequency of shortages than an increase in the variation in the total workload of an

15 Page of International Journal of Production Research order (VolVar). However, this holds only true for the heuristic approaches, as the optial solution is copletely insensitive for changes in these experiental factors. For each row in Table, 000 probles have been solved (0000 in total), and the nuber of probles that did result in a shortage in the optial solution was zero. Hence, expectation (x) cannot be confired. The perforance difference between the heuristic approaches depends on the settings of the experiental factors. Figure shows that a change in MixVar results in slightly different gradients of the curves that represent the five solution approaches. Hence, relative perforance of the heuristics if copared with each other changes if MixVar changes. However, (i), (iv), and (ix) do not hold, as the absolute perforance of all heuristics deteriorates for high MixVar, while expectation (iv) can be confired. And even for the relative perforance iproveent in average shortage, FillCap is better than the other heuristics. ***INSERT FIGURE ABOUT HERE*** The results show that the difference between heuristics and the optial solution is high. Figure shows that if MixVar is low, the ean frequency of shortages during a cycle is between 0% and % if the heuristics are being used, and increases to 0%-% if MixVar is high, while the optial solution approach is still able to find a solution without shortages. The frequency of shortages equals zero if the optial solution approach is used. The required CPU tie for obtaining the optial solution is less than 0 seconds, with an average of. seconds. Therefore, the use of the optial solution approach is a realistic alternative if the size of the proble is restricted and leads to substantial savings and an iproved capacity balance. Note that the size of probles considered in this study is realistic for any real life situations. Note also that in our experients the optial solution was insensitive to the value of the paraeter w t =0., as a trade off between short ter and long ter effects could be avoided (the optial solution was always able to avoid overtie). Table shows that VolVar has less effect on the heuristics. FillCap is indeed relatively insensitive to changes in VolVar, which confirs (ii), but AvgLoad is not perforing better for high VolVar, as expected in (v). Finally, we take a look at the effect of the paraeters n and. An increase in the nuber of periods per cycle decreases the perforance, and hence expectation (xi) is confired. However, in case of low ix variation, the effect is very sall. That eans that the average shortage per bucket decreases if n increases (xii). Apparently, longer cycles actually ake capacity balancing ore effective. With respect to an increase in, a strange phenoenon appears. The direction of the effect depends on the value of MixVar. If MixVar is low, an increase in deteriorates the perforance of the heuristics, but if MixVar is high, an increase in iproves the perforance of all heuristics. Therefore, expectation (xiii) is not confired. There is an effect for all heuristics if changes, but we have to accoodate for the interaction effect with MixVar. Note that Table already showed that this interaction is iportant, especially for explaining the variance in the shortage perforance easure. Conclusions The concept of synchronous anufacturing is behind several popular shop floor anageent approaches. It ais at achieving short and reliable throughput ties through the introduction of fixed transfer oents between several stages of production. This causes a loading of the resources that can be predicted in advance, which akes it easier for planners to do their job. If capacity requireents fluctuate over tie, planners ay have to enhance the order release decision in order to avoid costs of changing the total available capacity. Our study reveals that the quality of the order release decision is the key to an iproved capacity balance over the periods. In order to develop solution approaches for this ulti-product ulti-stage proble, we used insights fro ixed-odel assebly line sequencing algoriths and workload control literature. Four heuristics and a

16 International Journal of Production Research Page of atheatical odel were developed to solve the capacity balancing proble in a synchronous anufacturing syste. We tested the five solution approaches in a rolling schedule. This enabled us to take into account the consequences of the forer sequence decisions of the approaches for the new cycle. We evaluate the average total capacity shortage per cycle and the frequency of shortages for the various solution approaches. Our experiental design evaluates the perforance of the approaches for two levels of variation of total capacity requireents per order and for two levels of variation of the distribution of the workload over the stages. Moreover, we evaluated the effect of changing the length of the planning horizon and the nuber of stages. Surprisingly, the siple FillCap heuristic that selects an order that consues as uch capacity in the first stage of its release as possible perfors in all experients better than the ore sophisticated capacity balancing heuristics. This holds true even if the nuber of stages increases. We expect that the ain reasons are to be found in the bad balance at the end of the cycle, as the ore sophisticated heuristics do ai at a very good balance at the start of the cycle, without taking into consideration the effects of their choices at the end of the cycle. However, this study has not exained the ain reasons for the bad perforance of the sophisticated heuristics, as it expected the to perfor better. Future research should elaborate on this. The perforance of the heuristics is very sensitive to an increase in the variation of the distribution of workload over the stages. Increased variation of the total capacity requireents of an order has only a arginal ipact on perforance. The use of the atheatical odel instead of one of the heuristics results in an alost total eliination of shortages in our experients. Shortage frequencies of at least -0 percent were avoidable by using the atheatical odel. The tie needed for finding an optial solution is very short for the probles we have solved (up to orders and 0 stages). The size of probles that we have investigated is realistic for real life anufacturing systes, and ipleentation of the atheatical optiization odel is easy to accoplish. We conclude therefore that the use of such a solution approach should be considered, as it strongly iproves the capacity balance in synchronous anufacturing systes. Future research should verify if odifications of the proposed heuristics for synchronous anufacturing lead to perforance iproveents. We suggest to develop and test heuristics that take into account uncertainty with respect to the work content and available capacity. Genetic algoriths can be applied as well. The current study has applied deterinistic optiization in a rolling schedule horizon. We think that the use of a rolling schedule has iportant benefits when testing the perforance of solution approaches and should be applied to other probles as well. Acknowledgeents We like to express our thanks to Professor G.A. Süer, University of Ohio, for providing us any details on the jewellery fir and stiulating discussions. References Ahadi, R.H., and Wurgaft, H.,. Design for synchronized flow anufacturing. Manageent Science, 0 () -. Askin, R.G., and Standridge, C.R.,. Modeling and analysis of anufacturing systes. New York: John Wiley & Sons. Baker, K.R.,. An experiental study of the effectiveness of rolling schedules in production planning. Decision Sciences,, -.

17 Page of International Journal of Production Research Becker, C., and Scholl, A., 00. A survey on probles and ethods in generalized assebly line balancing. European Journal of Operational Research,,. Benders, J., and Riezebos, J., 00. Period batch control: classic, not outdated. Production Planning and Control. (), -0. Bukchin, J., and Masin, M., 00. Multi-objective design of tea oriented assebly systes. European Journal of Operational Research,, -. Burbidge, J.L.,. The principles of production control. Plyouth: MacDonald and Evans. Burbidge, J.L.,. Period Batch Control. Oxford: Oxford University Press, Clarendon Press. Cesani, V.I., and Steudel, H.J., 00. A study of labor assignent flexibility in cellular anufacturing systes. Coputers & Industrial Engineering, (), -. Cevikcan, E., Durusoglu, M.B., and Unal, M.E., 00. A tea-oriented design ethodology for ixed odel assebly systes. Coputers & Industrial Engineering, In Press. Available fro: DOI:0.0/j.cie Ding, F-Y., and Cheng, L.,. An effective ixed-odel assebly line sequencing heuristic for Just-In- Tie production systes. Journal of Operations Manageent,, -0. Gronalt, M., and Hartl, R.F., 00. Workforce planning and allocation for id-volue truck anufacturing: a case study. International Journal of Production Research, (), -. Karabati, S., and Sayin, S., 00. Assebly line balancing in a ixed-odel sequencing environent with synchronous transfers. European Journal of Operational Research,, -. Land, M.J., and Gaalan, G.J.C.,. Workload control concepts in job shops: a critical assessent. International Journal of Production Econoics, -, -. Lee, C-Y., and Vairaktarakis, G.L.,. Workforce planning in ixed odel assebly systes. Operations Research, (), -. Miltenburg, J.,. Level schedules for ixed-odel assebly lines in Just-In-Tie production syste. Manageent Science, (), -0. Morton, T.E., and Pentico, D.W.,. Heuristic scheduling systes with applications to production systes and project anageent. New York: John Wiley & Sons. Ouenniche, J., and Boctor, F.,. Sequencing, lot sizing and scheduling of several products in job shops: the coon cycle approach. International Journal of Production Research, (), -0. Riezebos, J., Korte, G.J., and Land, M.J., 00. Iproving a practical DBR buffering approach using Workload Control. International Journal of Production Research, (), -. Schenner, R.W., and Swink, M.L.,. On theory in operations anageent. Journal of Operations Manageent, () -. Scholl, A., and Becker, C., 00. State-of-the-art exact and heuristic solution procedures for siple assebly line balancing. European Journal of Operational Research,,. Stadler, H., 000. Iproved rolling schedules for the dynaic single-level lot-sizing proble. Manageent Science, (), -. Stevenson, M., Hendry, L.C., and Kingsan, B.G., 00. A review of production planning and control: the applicability of key concepts to the ake-to-order industry. International Journal of Production Research, (), -. Süer, G.A.,, Operation and control of cellular systes at Avon Loalinda, Puerto Rico. In: N.C. Suresh, and J.M. Kay, eds. Group technology and cellular anufacturing: state-of-the-art synthesis of research and practice. Boston: Kluwer Acadeic Publishers, -. Süer, G.A., and Gonzalez, W.,. Synchronization in anufacturing cells: a case study. International Journal of Manageent and Systes, (), -.