In bucket brigade" manufacturing, such as recently introduced to the apparel industry, a production line has n workers moving among m stations, where each w orker independently follows a simple rule that determines what to do next. Our analysis suggests and experiments con rm that if the workers are sequenced from slowest to fastest then, independently of the stations at which they begin, a stable partition of work will spontaneously emerge. Furthermore, the production rate will converge to a v alue that, for typical production lines, is the maximum possible among all ways of organizing the workers and stations. Key words: production line, line-balancing, self-organizing systems, discrete dynamical systems, xed p oint School of Industrial and Systems Engineering, Georgia Institute of Technology, A tlanta, Georgia 30332-0205 USA. E-mail: john.bartholdi@isye.gatech.edu y Graduate School of Business, The University of Chicago, Chicago, Illinois 60637 USA. E-mail: don.eisenstein@gsb.uchicago.edu 1 1 TSS production lines Traditional means of organizing a production line, such as a classical assembly line, are in exible. In a classical assembly line, workers are assigned xed work stations and the station with the greatest work content determines the production rate. Realistically, there are only two w ays to change the production rate: either change the number of shifts or else redistribute the tasks, tools, and parts over di erent stations. The rst allows only coarse adjustments and the second is expensive and disruptive. It is particularly important that production systems be exible when products have extreme seasonalities or short life-cycles, such as in the apparel industry. T o increase exibility of production in the apparel industry, a v ariation of the assembly line has recently been introduced in which there are fewer workers than stations and workers walk to adjacent stations to continue work on an item. Control of the line is decentralized: Each w orker independently follows a simple rule that determines what to do next. This idea has been commercialized by Aisin Seiki Co., Ltd., a subsidiary of Toyota, and named the Toyota Sewn Products Management System", or TSS 1 . TSS is used in the manufacture of many t ypes of sewn products, including apparel, furniture, shoes, hand bags, suitcases, and sh nets. Here is how TSS works. Call each instance of the product an item and consider a ow line in which each of a set of items requires processing on the same sequence of m work stations, as in Figure 1 . A station can process at most one item at a time, and exactly one worker is required to accomplish the processing. All items are identical and so each requires the same total processing time according to some work standard, which w e normalize to one time unit". Let the processing requirement at station j be p j , a xed percentage of the total standard work content of the product. 1 TSS" is a registered trademark of Aisin Seiki Co., Ltd. TSS is marketed in the western hemisphere by Americas 21st, Inc. A TSS line functions as a sort of bucket brigade" in which each w orker carries an item from station to station, processing it at each station, until passing it o to a subsequent w orker. This behavior can be realized by n umbering the workers from 1 to n according to their sequence on the line in the direction of product ow and requiring each w orker to independently follow this rule: TSS Rule forward part Remain devoted to a single item, and process it on successive w ork stations where at any station the worker of higher index has priority. If your item is taken over by y our successor or if you are the last worker and you complete processing the item, then relinquish the item and begin to follow the Backward Part. TSS Rule backward part Walk back and take o ver the item of your predecessor or, if you are the rst worker, pick u p r a w materials to start a new item. Begin to follow the Forward Part. Two points are worth emphasizing about the TSS rule. The rst is that a worker can be blocked during the forward phase if she 2 is ready to work at a station but that station is occupied by another worker. Under the TSS rule each w orker remains busy unless blocked, in which case she must wait for the station to become available. The second point is that during the backward phase each w orker interrupts her predecessor to take o ver her work. In e ective TSS lines a great deal of e ort 2 Nearly all the TSS workers we h a ve seen are female. This re ects the demographics of the apparel industry. 3 is invested to avoid introducing idle time during the hand-o ". For example, the line is con gured in a U-shape to reduce walking time, workers sew standing up at waist-high machines, and workers practice making the hand-o smoothly. A t ypical TSS line that we observed was devoted to the production of women's slacks. It had seven stations and was sta ed by three workers. The rst worker began sewing cut cloth; and the last worker ironed the slacks, attached labels, folded and packaged the slacks, and put them in a box for shipping. The total work content of the slacks was about seven minutes, with 45 90 seconds of work at each station. It was no more than 2 3 seconds from the time the last worker nished a pair of slacks and began walking back to take o ver from her predecessor until the rst worker began a new pair of slacks. In fact none of the TSS lines we h a ve observed formalized a TSS rule" nor did they follow our rule to the letter. Instead there have always been local improvisations to account for particulars of the product, the equipment, or the team members. Moreover the standard implementation of TSS has changed since its introduction to the US apparel industry in 1989. The TSS rule" is our abstraction, which captures the essential behavior of TSS lines. Many questions leap to mind: How does a TSS line behave? Is TSS e ective? How does one control or even predict the production rate? To be sure, factory managers are discovering answers to these questions on the shop oor. Here we begin a formal analysis of a model of TSS. Our results suggest and experiment con rms that if TSS workers are sequenced from slowest to fastest then, during the natural operation of the line, the work content of the product will be spontaneously reallocated among the workers to balance the line|without conscious intention by the workers and without intervention by management. This capacity for self-organization allows management to ne-tune the production rate by simply changing the numberof workers on the line, which in turn elicits a spontaneous reallocation of work. The standard work content of the product is represented as a line segment normalized to length 1, which is partitioned into intervals corresponding to the work stations. The position of worker i is given by x i , the cumulative fraction of work content completed on her item. It is di cult to visualize the ow of items on a TSS line because the workers move asynchronously and are not explicitly limited to any particular set of stations. However, we can view the line as a dynamical system Devaney, 1989 by expressing the position of worker i as the fraction x i of work completed on her item, as illustrated in Figure 2 ; then the state of the system at any time can be summarized by the vector of worker positions x = x 1 ; : : : ; x n . The phase space of the system is a subset of the closed n-cell f x 1 ; : : : ; x n : 0 x 1 x n 1 g as illustrated in Figure 3 . The inequalities 0 x 1 x n 1 arise because the TSS rule does not allow w orkers to pass one another. We model each w orker i by a v elocity function v i x that gives her instantaneous work velocity at position x 2 0; 1 when not blocked by an occupied station. To a void modeling pathologies we require that Each v i is continuous almost everywhere on 0; 1 ; and There exist numbers b and B such that, for each w orker i, 0 b v i x B 1 for all x 2 0; 1 . Roughly speaking, these restrictions say that a worker cannot abruptly change speed at every instant; and workers are neither in nitely fast nor in nitessimally slow. Our model of worker skills includes one used by the apparel industry, wherein each w orker has a documented skill pro le giving her velocities at di erent tasks The phase space of a TSS line with two w orkers, whose positions are given by x 1 ; x 2 , is that portion of the upper triangle outlined in bold. The feasible region is above the dotted line because worker 1 can never pass worker 2. The tick marks on the axes correspond to the partition of work among the stations and the saw-toothed edge of the phase space arises because no more than one worker can use a station at a time. 6 as a percentage of work standards. This corresponds to a v i that is a step function. The TSS lines we h a ve seen required no more than a few seconds for any worker to walk back and take o ver the work of her predecessor. Therefore we make the modeling assumption that, while the workers move forward with nite velocities, they move backward with e ectively in nite velocity, so that when the last worker nishes an item, then|at the same instant|worker n takes over from worker n , 1, who takes over from worker n , 2, . . . , who takes over from worker 1, who introduces a new item into the system. We s a y that the line resets at such an instant. This simpli cation frees us from worry about the details of the continuous-time evolution of the TSS line; instead we can restrict our attention to the sequence x 0 ; x 1 ; x 2 ; : : : ; x t ; : : : of worker positions at those instants immediately after the line resets 3 . Note that x t 1 = 0 and for convenience we de ne x t n+1 = 1 . In the terminology of dynamical systems, x t is the t-th iterate of the TSS system and the sequence of worker positions is the orbit beginning at x 0 . Let f be the function, de ned implicitly by the TSS rule, that maps the vector of worker positions after one reset to that after the subsequent reset, so that x t+1 = fx t . Now w e can study the behavior of a TSS line by studying its orbits, where each orbit x t = f t x 0 1 t=0 is determined by the initial positions x 0 of the workers. 3 Tracking the behavior of a TSS line 3.1 Repeatable behavior A TSS production line is balanced" if each w orker repeats the same interval of work content on successive items. A balanced line produces at a steady rate; 3 Such a subset of the phase space is called a Poincar e s e ction. Restricting the phase space in this manner is a standard technique for analyzing dynamical systems Morrison, 1991. 7 and each w orker can concentrate on a subset of the work content. Because the TSS rule imposes no apparent restrictions on where a worker might m o ve, it is not clear whether a line can maintain balance or indeed whether balance can be achieved at all. Our rst result shows that balance is always at least theoretically possible; that is, there exist worker positions such that, if the workers begin at these positions, then, after completion of each item, they will reset to exactly these same positions to begin work on the subsequent item. Theorem 1 For any TSS line, there exists a xed p oint x = fx ; that is, there exist worker positions x such that if the workers start at positions x , then they will always reset to x . Proof Appendix, section A.1. One of the main concerns of this paper will be to determine how a TSS line can be designed to operate at or near its point of balance that is, its xed point so that the workers repeat the same portions of work. We s a y that worker j is faster than worker i if she is discernibly faster at every portion of work content: sup We rst observe that, if the TSS workers are sequenced from slowest to fastest, then the xed point is unique. In contrast, when workers are not sequenced from slowest to fastest, then there can be multiple xed points. 4 This condition is slightly stronger than the simpler requirement that v i x v j x but is necessary to avoid technical pathologies in which the velocity of one worker approaches that of the other in the limit. Lemma 1 If the workers are s e quenced f r om slowest to fastest, then the xed point of the TSS line is unique. Proof Appendix, section A.3. Our main result is that, if the workers are sequenced from slowest to fastest, then there exists a unique xed point to which all orbits converge, so that the behavior of the line is independent of the starting positions of the workers. In addition, if there is no blocking, then each w orker invests the same clock time in each item produced. It makes sense to put slower workers before faster ones to avoid blocking a local phenomenon; what may be surprising is that this is su cient to elicit global organization: the spontaneous emergence of balance.  An interesting special case of our model illustrates the essential analysis and probably provides an adequate description of most real TSS lines. Assume that each w orker i has a velocity that is constant o ver the entire unit of work, so that v i x = v i . Since all velocities are constant, there are no advantages to specialization of labor and so the largest possible production rate under any organization of the workers is from which it follows by simple algebra that for i = 2 ; : : : ; n , which m a y b e i n terpreted as the clock time separating workers i and i+1 at the start of iteration t + 1 , w e can summarize the dynamics of the line as follows: Rewriting these equations as a linear system a t+1 = Ta t ; we observe that this sequence of iterates converges because T is the transition matrix of a nite state Markov c hain that is irreducible and aperiodic Resnick, This may b e i n terpreted as showing that, to con gure a bucket brigade from well to re, one should put the fastest people close to the re; then the people will, without intention, space themselves to convey the greatest possible ow o f water upon the re. The system optimizes itself. The details of this proof illustrate the general ideas of the full argument. The main trick is to simplify analysis by looking, not directly at the positions of the workers, but at the clock time a t i required for each w orker i to reach the position of her successor. We refer to this time as the allocation of work suggested by the partition x t . Allocations are de ned for the general model in the appendix, section A.2. When workers are sequenced from slowest to fastest, the largest allocation|which w ould be the cycle time of the line if the allocations suggested by the current x t were xed|converges from above and is guaranteed to have decreased after each completion of n items. Thus the production rate increases to a limit whose value is independent of the starting positions of the workers. Figures 4 and 5 show the convergence of a system from two complementary points of view. Figure 4 shows an example of how the movement of the workers stabilizes, with the faster workers eventually allocated more work; and Figure 5 shows the convergence of the system within the state space of worker positions. These simulations were generated by three workers of constant v elocities v = 1; 2; 3. In computational experiments with workers sequenced other than from slowest to fastest, our model of a TSS line can fail to balance itself. By The positions of the workers on the production line at successive instants when it resets. Since the position of the rst worker is always 0 when the line resets, only x 2 ; x 3 , the positions of the second and third workers, are plotted here. From any initial position the system converges to the xed point 1 =6; 1=2. The production rate also converges to a unique value|in this instance, 6, the maximum possible|that is independent of where the workers start on the line. there always exists a balance point so that workers always reset to the same position; the trouble is that the xed point can be a repeller, so that if the system ever deviates, however slightly, from that point, then the system must inexorably diverge from it Devaney, 1989. Typically the line becomes trapped in periodic behavior: It sputters", producing erratically and at suboptimal production rate. Figure 6 shows a simulation, generated by w orkers of constant v elocities v = 3 ; 1; 2, in which the xed point is a repeller and any orbit that strays must eventually be trapped by a limit cycle with production rate less than that at the xed point. In the limit cycle a faster worker is repeatedly blocked by a slower worker, with consequent w aste of productive capacity. In other simulations we h a ve found instances of quite large cycles, some at the limits of the numerical resolution of our computer and of the patience of the observers. Our model is also capable of quasi-periodic" behavior, which means that it is predictable but not periodic. For example, with constant v elocities v = 2 ; 1; 2, all orbits converge to the periphery of an ellipse, the center of which is a xed point|but worker positions never repeat. For a xed set and sequence of workers our simulations have shown such phenomena as multiple xed points, both attractors and repellers, multiple limit cycles, and long-term behavior that depends on the starting positions of the workers. This suggests that, if workers are not sequenced from slowest to fastest, there can be a structural tendency toward persistent i m balance in a TSS line. This could be a practical problem if the imbalance is signi cant. In a companion paper we h a ve catalogued all possible asymptotic behavior of 2-and 3-worker lines and interpreted its signi cance for practice Bartholdi, Bunimovich, and Eisenstein, 1995a. More troubling than complicated behavior is anomalous behavior. The simplest manifestation of this is that adding a worker to the line can decrease the production rate if the slowest-to-fastest sequence is not respected. For example, Figure 6 : The positions of the workers on the production line at successive instants when it resets. In this instance the xed point 1 =2; 2=3, indicated by , a c hieves the largest possible rate of production, but is a repeller, and any orbit that strays from it will be trapped by the attracting but suboptimal limit cycle consisting of the points 3=15; 7=15, 7=15; 7=15, 11=15; 11=15, and 2=5; 13=15, indicated by 's. As the system is trapped by the limit cycle, the cycle time of the line oscillates and the average production rate converges to 60=11, which is less than the optimum value of 6. consider an m station line with processing times p = 1= m ; : : : ; 1=m sta ed by a single worker of velocity k, where 1 k m. The production rate of this line will plummet from k items per time unit to only a little more than 2 items per time unit if a worker of velocity 1 is added to the end of the line, where she repeatedly blocks the faster worker. Thus one can induce an arbitrarily large gap between production capacity and realized production rate. It is less obvious that increasing the velocity o f a w orker can decrease the production rate. For example consider a TSS line with processing times p = 1=2; 1=4; 1=4. On this line workers of constant v elocity v = 2; 1; 1 will achieve a production rate of 4 items per unit time; but if worker 3 doubles her velocity, the rst worker will always be blocked and the production rate will decrease from 4 to 8=3. Therefore, increasing production capacity b y 25 causes a 33 decrease in realized production rate! We emphasize that system behavior can be neither complicated nor anomalous when workers are sequenced from slowest to fastest. Theorem 4 If workers on a TSS line are maintained i n s e quence f r om slowest to fastest, then adding or speeding up a worker will never decrease the production rate. Proof Appendix, section A.3. Here we demonstrate that the logic of TSS cannot by itself guarantee the best production rate if there is a pathological mismatch b e t ween the sequence of workers and the assignment o f w ork content to stations. Nevertheless a TSS line in which w orkers are sequenced from slowest to fastest will always achieve a production rate that is good in the following sense: Other sequences can perform much w orse but not too much better. The following example shows that a sequence other than slowest-to-fastest can be arbitrarily less productive than the slowest-to-fastest sequence of workers. Consider a TSS line with p = ; 1 , and two w orkers, one of constant v elocity and the other of constant v elocity 1 , . In this sequence the workers achieve a production rate of one item per time unit; but reversing the sequence gives a production rate of =1 , items per time unit, which can be made arbitrarily small. Thus the worst-case ratio of production rates is unbounded above. On the other hand, a sequence of workers other than slowest-to-fastest cannot achieve a production rate that is too much" better than that of the slowestto-fastest sequence. More speci cally, when the fastest worker is last, the production rate of a TSS line at its xed point is always within a factor n of the best achievable by a n y other sequence of the workers. This follows because the fastest, last worker is never blocked. One can construct examples that achieve this bound asymptotically. We emphasize that, although this worst-case behavior is possible within our model, it is not a practical problem because neither persistent blocking nor pathological mismatch o f w orkers to stations will be found. First, persistent blocking is not tolerated. If some station is recognized as a bottleneck, then either workers are removed from the line or else the station is duplicated and all of our results can be shown to hold for a line with parallel stations. Second, a pathological mismatch o f w orkers to stations is generally not possible where, as in the apparel industry, most tasks are variations of a single skill such a s simple dexterity. Theorem 3 therefore suggests that real TSS lines will achieve a nearly maximum rate of production if the workers are sequenced from slowest to fastest.  The details of our model are con rmed by evidence, both anecdotal and experimental, and by industry practice. More importantly, the predictions of our model are con rmed by the operation of real lines, in both the laboratory and at commercial sites. Here are the distinctive features of our model and its predictions. We timed several workers over hundreds of task executions on the shop oor at a commercial site. Figure 7 shows the distribution of actual times for a speci c worker to complete a speci c task. It is representative of all the workers we timed and it is consistent with measurements by factory personnel. In analyzing actual processing times measured on the shop oor, we discerned two sources of variance in the execution of a given task by a given worker. The rst source was the inevitable small noise". This was generally because small variations in the cut cloth resulted in small variations in the time to position the cloth under the needle. While this component of task time seems 18 properly described as random", its variance was insigni cant. It is re ected in the rst, major peak in the distribution of task times in Figure 7 . The second source of variance was that due to occasional interruptions to which sewing is subject and which form the second, small peak in Figure 7 . Most of the long delays were not randomly occurring: Instead, these were the regular pauses to position new bundles of raw materials. Signi cant random" interruptions, such as a dropped garment, a quality problem, or a thread break, occurred much less frequently. We judged the rst type of variance to be insigni cant because, ignoring documented pauses to position raw materials, the coe cient o f v ariation of the processing time was less than 10 for an average worker and task. This is to be expected because the workers perform the same set of tasks several hundred times each d a y and so become quite consistent at it. Accordingly we modeled processing times as deterministic. Interruptions are not intrinsic to our model; instead, we consider them to be extraneous shocks to the system. Our model explains how the dynamics compensate for such shocks to re establish balance. We h a ve found contexts in which a stochastic model seems appropriate. For example, Bartholdi, Bunimovitch, and Eisenstein 1995b analyze TSS-style picking from a owrack i n a w arehouse supporting retail operations. In this case the composition of orders to be picked can vary signi cantly and seems best modeled as a random variable. Su ce it to say here that under mild assumptions the vector of worker positions converges to a random variable, the distribution of which is independent of the starting positions of the workers. Workers vary signi cantly in speed, as shown in Figure 8 , which gives the average velocities of sixty-one apparel workers at a commercial site. Furthermore, this large variation in worker speeds persists because the turnover rate among employees in the US apparel industry is over 40 per year, with many new 19 0.5 0.6 0.7 0.8 0.9 1.0 1
REF show that if workers are sequenced from slowest to fastest in a production line and if they have the possibility of changing stations, but not of passing another worker, a stable partition of work will spontaneously emerge.
A Production Line That Balances Itself
Computer Science Mathematics