Synergy, Collaboration and Self-Balancing in Production Systems - The Bucket Brigades

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SYNERGY, COLLABORATION AND SELF-BALANCING IN PRODUCTION SYSTEMS - THE BUCKET BRIGADES Antoneta luliana BRA TCU* and Alexandre DOLGUl** *Advanced Control Systems Research Centre "Duniirea de Jos" University ofGala/i 111. Domneascii. 800201 Gala/i. Romania ** Centre SIMMO Ecole des Mines de Saint Etienne 158. Cours Fauriel. 42023 Saint Etienne cedex 2. France E-mail: [email protected]@emse.fr

Abstract: The line balancing problem, which is NP-hard, concerns any industrial flow line activity. The "bucket brigades" represent a production line example of intrinsic balancing, realized by the functioning protocol itself. The global optimality results from a decentralized control strategy, based upon the collaboration and the synergy of many individual (human) "agents". The theoretical analysis of the bucket brigades has started with the "Normative Model". This paper intends a brief survey of the analysis approaches based upon the extensions of this model and the generalizations of the bucket brigades principle. Copyright © 2003 IFAC Keywords: production systems, work organization, agile manufacturing, selforganizing systems, dynamic systems.

line, thus reducing the design effort. This property expresses the self-organizing capacity of the bucket brigades. There is not anymore a strict assignment of equipment and tasks to workstations; the well-known concept of "workstation" is given a new meaning, as the equipment are, in fact, human operators, which are not strictly assigned to certain workstations, but can move among them.

I. WHAT ARE THE BUCKET BRIGADES? Flow manufacturing lines can be found wherever "products" may be imagined to move along, from worker to worker, for example, as in an assembly line: products are progressively assembled as they move down the line towards completion. In the classical approach, the station with the greatest work content determines the production rate of the line. For the flow lines, the problem is the workloads balancing, such to maximize the production rate.

1.1 The Normative Model

The bucket brigades operate as follows: each worker carries an item towards completion; when the last worker finishes his item, he sends it off and then walks back upstream to take over the work of his predecessor, who walks back and takes over the work of his predecessor and so on, until, after relinquishing his item, the first worker walks back to the start to begin a new item. A worker might catch up to his successor and be blocked from proceeding. The TSS Rule requires that the blocked worker remain idle until the station is available. Only the last worker is never blocked and he determines the line productivity.

The bucket brigades represent an alternative way of work organization of flow line activities. The bucket brigades use human operators (named "workers"), fewer than the workstations, which are allowed to move between adjacent stations. All the operators respect the same rule, named "TSS Rule" (Toyota Sewn Management System, registered trademark of Aisin Seiki Co. Ltd.) When following the TSS Rule, a flow line is spontaneously maximally productive and autonomously maintains its optimal production rate, without any conscious intervention. The selfbalancing emerges like an intrinsic property of the

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functioning rule, simple and identical for all the workers, and the work-in-process is only that in hands of workers, because the input of a new item is ordered by the output of the previous one. Such a line illustrates the concept of "pull system", both globally and locally too. The transport and the manipulation of items are eased, as the workers carry the items themselves. Due to the self-balancing property, the task durations need not to be exactly measured; furthermore, the line responds spontaneously and instantaneously to changes of configuration. The production rate can be finely tuned by simply adding or freeing up workers on/from the line.

Let m be the number of workstations and n be the number of workers. The Normative Model is the simplest model of the bucket brigades dynamics and it is based upon the following assumptions (Bartholdi and Eisenstein, I 996b ): a) total ordering of workers by velocity: each worker i is modelled by a distinct velocity function, v,{x) , giving his instantaneous work velocity at position x E [0; I); it is said that worker j is faster that worker i (Vi-
b) insignificant walk-back time: workers move backward infinitely faster than they work; c) "smoothness" and predictability of work: the total standard work content required by an item, which is normalized to 1, is continuously and uniformly spread along the line and partitioned into intervals corresponding to workstations (see figure I). X,

• 1

0

--+.~X, ) I

PI

xz --+·~x)

1

Therefore, it is not necessary to conceive a new control system, because the bucket brigades principle assures that the control be embedded into the system itself. Such lines illustrate the concept of "lean" manufacturing.

...

The global optimality of this kind of production lines results from a decentralized control strategy, based upon the collaboration and the synergy of many individual (human) "agents". Their study is required by the necessity of using humans in some special production environments. Bartholdi et al. (1995) have identified that the bucket brigades are applicable especially where: - all the operations of the line require almost the same work ability; - the workers can easily move between the workstations; - the equipment placed in the stations is more cheaper than the labour cost; - there are important demand fluctuations of the products.

I

pz

p.

Fig. 1. The standard work content split into workstations, the workers velocities and positions

The vector ,!(t)=[XI(t) X2(t) ... xn(t)f represents the system' s state at time t. An iteration is the time elapsed between two successive handoff moments (also called reset moments). The sequence {l°>, ,!(I), ,!(2) , ... ,!(k), ... } denotes the successive reset positions (XI(k)=O). Let/be the function that ma}!s the vector of workers reset positions so that ,!( I)=j(,!(k». The orbits {,!(k+')=fk{,!(O»hz()" ,... , determined by the initial conditions ,!(O), describe the line behaviour. The stationary behaviour is defined by the fIXed points of f, which are points of balancing.

In the rest of the paper it is intended to briefly survey some new trends in the analysis of the bucket brigades as example of agile manufacturing.

In the literature there are reported different analysis approaches of the self-balancing lines. The most important results were obtained by using the theory of stochastic dynamic systems (Bartholdi and Eisenstein, 1996b), where there were stated the quite realistic assumptions of the Normative Model. This model has become a guide for the bucket brigades behaviour; it was used, for instance, in the analysis of bucket brigades as nonlinear and hybrid dynamic systems (Bratcu and Minzu, 1999; Bratcu, 200 I). Indeed, the predictions of this model were validated in practice (Bartholdi and Eisenstein, 1996a; Muiioz and Villalobos, 2002).

2. SOME SYNERGIC MODELS OF THE BUCKET BRIGADES 2.1 Analysis

The bucket brigades constitute a particular class of systems with collective synergy. The management of such systems is approached by methods of the dynamic systems theory, oriented rather to the deterministic point of view. The advantages of this latter class of methods have been emphasized by Bunimovich (200 1), rediscussing the behaviour of bucket brigades from a more general angle. It is defined a dynamic system generated by the movement of N particles in a segment [O,L), with velocities generally depending of their positions within the segment, v,{x), i= 1,2, ... N. The case where the particles are allowed to pass one another was called "Qne ~ay ~treet" - OWS, whereas the contrary case was named "Qne ~ay tlarrow ~treet" - OWNS.

1.2 Main advantages versus the traditional cases

The self-balancing property eliminates the necessity of solving a NP-hard classic assembly line balancing (ALB) problem, as the total cycle time is implicitly minimized. The main advantages of the bucket brigades are the increase of the flexibility and the reduction of the work-in-process. Indeed, the line is governed by a

The two cases of dynamic system can describe

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various situations encountered in the practice: the goods travel in warehousing, the items forwarding along a production line, the travel of flexible workers (as in the bucket brigades, for example). The problem to solve is how "to organize" such a system in order to maximize its productivity. In the second case - where the bucket brigades belong to - this means to minimize the effect of blockings.

optimally scheduling the productive elements workers, robots, etc. - on the line. Bunimovich (1999) listed the main phases of a solving algorithm for this problem: I) determine all the sequences having a stable fIXed point (stable stationary state); 2) choose among the sequences found in the first phase the one which is the most consistent with a production line configuration.

Supposing the finiteness of velocities in all the points of the segment [O,L], the existence and the uniqueness of the fIXed point - which describes the optimal dynamics, the self-balancing - of a system of bucket brigade type are guaranteed by the next three results.

This algorithm represents a particular case of the general approach of S.tabilization of a larget Regime - STR, issued from the theory of automatic control. In bucket brigades, the undesirable regimes are the blockings, because they perturb the dynamics and eventually introduce chaotic phenomena. The bucket brigades optimization is thus related to the chaos control theory (Bunimovich, 1999), where the goal is to find a regular (periodic) orbit and to stabilize it, by applying to the system a sequence of small perturbations (Chen, 2000).

Lemma (existence of a balancing point): The point

x· = [0

x; ... x~

(X;)2

r'

where:

(. .f

(L-x~t

x,

L

xl -x2

x,

( vJ(x)dx 0

3. EXTENSIONS OF THE NORMATlVE MODEL

r vN(x)dx

r v1(x)dx

X

(I)

The practice has shown the efficiency of the bucket brigades in relation to the traditionally balanced production lines, even under real conditions quite far from those supposed in the Normative Model. The bucket brigades are robust at large variations in relation with the conditions considered as ideal. This section is dedicated to some directions of extension of the theoretical analysis - from which the main is to relax the assumptions of the Normative Model and to generalizations of the bucket brigades' principle to production systems with other than serial spatial organization.

xN

z

is a fIXed point of a system of bucket brigade type. Theorem I (uniqueness of a balancing point): A fixed point given by relation (I) is the only fIXed point of a system of bucket brigade type. The fixed point verifying relation (I) can be stable, unstable or neutral (in the sense of the systemic concept of "stability"). The case of constant velocities was called Basic Model. In this case, the fixed point has a simpler expression, that is

[

0,

{x; (~\ If Vi )} =L.

,=1

,- 1

3.1 The case ofstochastic operating times

]T , reflecting the k =2,3•... N

In Bartholdi et al. (2001) it is relaxed the assumption of the uniform distribution - "smoothness" and predictability - of the work content. Thus, the third assumption of the Normative Model is replaced by the following one: the standard work content at each station is considered exponentially distributed of mean normalized to I . This assumption is equivalent to consider that the time required to worker i to accomplish a task is exponentially distributed of mean \lVi. The study of this case is done in analogy with the deterministic one. The authors have considered the continuous dynamics of a bucket brigade, including that in between the successive resets, for both of cases, deterministic and stochastic. They have theoretically proved (by convergence analysis in topologic spaces) that, more the stations' number increases, more the stochastic dynamics is closer to the deterministic one, predicted by the Normative Model. Moreover, this resemblance asserts by itself very quickly. Their results have been verified by both simulation, and by practical implementation in a national warehousing centre, Revco Drug Stores, Inc. Indeed, if the well ordering of workers does not change, the behaviour in the stochastic case is not sensibly different from that of deterministic operating times, except if the number of workers almost equals the number of

N

maximal production rate,

I

vi. The next theorem

i=1

gives the relation between the optimality of the Basic Model - the "self-balancing" - and the stability of its fixed point. Theorem 2 (necessary and sufficient condition of optimality): A sequence of particles of the Basic N

Model provides the maximal productivity,

I

Vi' if

i=1

and only if the correspondingflXed point is stable. This theorem shows that it can exist many sequences of particles leading to the maximal production rate (from which one is, as known, the increasing order of their velocities). But in the case of bucket brigades, the theorem is not valuable, because not all the sequences having a stable fIXed point are optimal (as simulations have shown; see, for example, Bratcu and Minzu, 1999).

2.2 Control issues The general problem of optimally organzzmg the manufacturing lines of OWNS type is reduced at

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workstations. Therefore. the Normative Model can be used as a guide of bucket brigades' behaviour. even for a quite realistic (small) number of workstations per worker and for a quite non-realistic (large) variance of the work content at each station. Nevertheless. the optimality of the slowest-to-fastest ordering has not been yet proved in the stochastic case. with 'i

There exist also other stochastic modelling approaches of the work sharing on a production line. in environments similar to bucket brigades. which consider identical workers (standard workers) (Bischak. 1995; Zavadlav et al.. 1996). This assumption seems quite far from reality. since it is the case of human beings. but it can provide an easy computable lower bound of the possible productivity in the case of heterogeneous workers. that is m/(n + m -I) (see the notations above). Bischak (1995) obtained the same result based on the equivalence of bucket brigades with cyclic queues. since the workers are identical. According to the last relation. the production rate can be judged upon the ratio mIn. that is how many workstations result averagely per worker. This relation suggests that it is preferable to have few workers with almost the same work velocity and that the number of stations be sufficiently superior to the number of workers.

=(..!..+..!..)/(..!..+~). i =1.2•.. .n-1 . Vb

Vn

Vb

Vi

3.3 Bucket brigades in non-serial configuration

The relaxations brought to different assumptions of the Normative Model have shown that the well ordering condition remains sufficient for the selfbalancing of a bucket brigade. even in cases quite far from the ideal one. corresponding to this model. Another question is whether the efficiency of the bucket brigades' principle remains unchanged in systems having spatial structures more complex than serial and which modifications and/or adaptations would eventually be necessary to the original functioning rules.

For a "sufficiently" variable work content. the "fixed" protocol of the bucket brigades can be in principle replaced by an adaptive. "real time" policy. which must allow in each moment the re-ordering of workers. To determine what sequence of workers is optimal at each moment is an interesting theoretical control problem. But its practical utility is weak. as it is probable to introduce transients required by the workers' re-adaptation. Moreover. the workers need to know the most clearly possible what to do next. Fig. 2. In-tree spatial structure of an assembly system: each arrow denotes a serial assembly sub-line; each node has the label of the emerging sub-line (Bartholdi et al.. 2003)

3.2 The case offinite backward velocity

The second assumption of the Normative Model the backward walk. infinitely faster than the working walk of workers - seems the more realistic and incontestable. But its relaxation is not less interesting. Thus. instead of an infinite backward velocity. Bratcu and Dolgui (2003) have considered a finite backward velocity. the same for all the workers. denoted by Vb. As a consequence. the reset is not instantaneous or simultaneous anymore. but it propagates from the last to the first worker. In the cited work. it is shown that placing the workers in increasing order of their work abilities remains also in this case sufficient for the line self-balancing. In the discrete dynamic systems theory. the analytic method is illustrated by simulation on a nonlinear model. Intuitively it is clear that the ideal case - with infinite backward velocity. as supposed in the Normative Model - is a lower bound of cases having a finite backward velocity. from the point of view of the line's productivity. This is proven by the expressions of the fixed point and of the production rate (Bratcu and Dolgui. 2003):

Bartholdi et al. (2003) have studied the extension of the bucket brigades' protocol for spatial configurations which are not serial anymore. but intree. The general case of an assembly system. formed by several assembly sub-lines. has been considered. On each such sub-line it is produced a single sub-product; the sub-products are next successively assembled to obtain the final product. Here it must consider the presence of buffers. considered to be placed at the end of each sub-line. Such a system can be figured as an oriented connected graph (see figure 2). where from each node emerges a single arrow. except from the root (from which does not emerge anyone). Each arrow is associated to an assembly sub-line. This graph represents in fact the precedence relation - partial order relation - between the tasks to assembly the sub-products. The first and the third of the Normative Model's

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assumptions are considered true for each serial subline: the total ordering of workers by their work abilities and the smoothness and predictability of work.

order, as it was the case of a single serial line. Differently from the serial case, the productive travel of a worker can be blocked if he cannot continue because of the sub-product's absence (the corresponding buffer is empty). Thus, to control such situations, the following protocol extension is proposed: if a worker is blocked because of the absence of sub-product(s) in the corresponding buffer(s), then he must put the just finished subproduct in its buffer and go backward to take the item of his predecessor (the next slower worker). In the cited work, the problem of finding a total order responding to the two optimization criteria is shown to be NP-hard and solved by a polynomial heuristic.

The basic idea for the adaptation of the bucket brigades to non-serial configurations is to convert the work distributed on the spatial graph in a work sequence equivalent to a serial line. This was called by the authors "one dimension reduction". It consists in finding a total order (of the sub-lines) consistent with the partial order described by the graph. For example, the order suggested by the numbering of figure 2 - G, C, B, A, F, I, E, D, H, J, K - meets the above-mentioned condition. There exist many such possible orders, so the problem is to optimize the choice. Two optimization criteria were defined to build the preferred set of total orders: I) to ensure the coherent progress of sub-tasks; 2) to minimize the duration of workers' movements. Many types of movement (travel) can be distinguished, since the choreography of workers is more complex than in the serial case: the forward travel, formed at its turn by the productive travel (with the work velocity) and the non-productive travel, between the end of a subproduct and the start of the next one, and the backward travel, entirely non-productive (considering that this is done along the shortest path between two nodes of the graph).

The authors have stated a lemma, which establishes a superior bound of the work-in-process of a system with the described configuration, that is 2·n·(d"".,.-I), where n is the number of workers and dmax is the maximal in-degree of the graph. 4. OPTIMAL ORGANIZING OF FLEXIBLE WORK FORCE It is useful to place the bucket brigades in a larger context, of production environments based upon mass synergy and cooperative work organizing. This research direction has been recently exploited, leading to more general models (Bunimovich, 2001), or has revealed some organizational aspects by comparison (Van Oyen et al., 2001).

In the serial case, the distinction between theforward and the backward travel was relevant because of the coincidence with the distinction between productive and non-productive. In this case, the forward travel of workers contains non-productive parts. Therefore, the second assumption of the Normative Model was reformulated as follows : the non-productive travel is much quicker than the productive one.

The opportunity of the workforce flexibility (agility) in the production systems was discussed in Van Oyen et al. (200 I). The authors formulate optimal policies for organizing teams of full cross-trained workers (who can perform any operation) in systems of "make-to-order" type both collaborative and noncollaborative. The collaborative systems of "maketo-stocK' type with constant ~ork-in-Rrocess (CONWIP) are analysed as well. By the way, the bucket brigades are such kind of systems.

For minimizing the travel of workers it is therefore sufficient to consider only the forward travel. Once the total order has been established, the bucket brigades' protocol can be executed following this

Table I Synthesis of oRtimal oolicies for the Rroduction systems with flexible workfOrce (Van Oyen et al.. 2001) Policy

Advantages

COLLABORATIVE optimal: E~pedite folicy (XP) - all the workers perform simultaneously the same task -

dynamical balancing; remains optimal in the presence of finite buffers between stations

NON COLLABORATIVE CONWIP sub-optimal: fick-and-Run (PR) optimal: XP each worker performs independently (but it is not the tasks following the FlFO protocol, t .::in:.:te.:.;rruc:..::oP"'t"'io::.:n"--_ _ _ _ _ __ the only one) -"w;.:.ith=o:..::uc:.. dynamical balancing; superior to bucket brigades protocol, if disposing of quite complex equipments; easier applicability than that of XP

Taking into account the meanings of terms used in the cited work, that is: "collaborative", when many workers can perform the same task, without loss of efficiency, and "non-collaborative", when it is required exactly one worker per task, one can note that the bucket brigades are non-collaborative systems. Their efficiency can be overtaken by other strategies, if disposing of quite ample equipments, as shown in table I, which resumes the different optimal policies.

5. CONCLUSION AND FUTURE WORK In relation with the traditionally balanced lines, the bucket brigades do not need but a simple constraint to work well: workers' sequencing in increasing order of their work velocities. Therefore, the design effort is notably reduced, if having correct data on the workers' performances. Moreover, the adaptability and the robustness of the bucket brigades - in relation with different types of

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Lines. To appear in European Journal of Operations Research. Bischak, D.P. (1995). Performance of a Manufacturing Module with Moving Workers. liE Transactions, 28/9, 723-733 . Bratcu, A. and V. Minzu (1999). A Simulation Study of Self-balancing Production Lines. In: Proceedings of the 1999 IEEE International Conference on Intelligent Engineering Systems INES '99, November 1-3, 1999, Stara Lesmi, Slovakia, 165-170. Bratcu, A. (200 1). Determination systematique des graphes de precedence et equilibrage des lignes d'assemblage. PhD Thesis, Universite de Franche-Comte, Besan~on, France. Bratcu, A. and A. Dolgui (2003). Une relaxation du Modele Normatif des lignes de production autoequilibrees (<< bucket brigades »). Actes de la 4eme Conference Francophone de Modelisation et Simulation - MOSIM '03, Toulouse, France, 23-25 avril 2003, Y. Dallery, J.-C. Hennet, P. Lopez (Eds.), SCS European Publishing House, 2003, 262-269. Bunimovich, L.A. (1999). Controlling production lines. In: Handbook of Chaos Control (H. Schuster (Ed.», 337-354. Wiley-VCH. Bunimovich, L. (2001). Dynamical Systems and Operations Research: A Basic Model. Discrete and Continuous Dynamical Systems, 112, 209218. Chen, G. (Editor) (2000). Controlling Chaos and Biffurcations in Engineering Systems. CRC Press LLC. Munoz, L.F. and J.R. Villalobos (2002). Work Allocation Strategies for Serial Assembly Lines under High Labor Turnover. International Journal of Production Research, 40/8, 18351852. Van Oyen, M.P., E.G. Senturk-Gel and WJ. Hopp (2001). Performance Opportunity of Workforce Agility in Collaborative and Noncollaborative Work Systems. liE Transactions, 33/9, 761-777. Zavadlav, E., lO. McClain and L.J. Thomas (1996). Self-Buffering, Self-Balancing, Self-Flushing Production Lines. Management Science, 42/8, 1151-1164.

changing the initial conditions (stochastic variability of the operating times, changing of the product type, modifications of the spatial configuration, etc.) shown by both simulation, and in practice too, and confirmed by theoretical analysis, are important strong points for their effective use on real lines. Indeed, the bucket brigades exist now mainly in logistics and warehousing and in manufacturing. In this latter case, the assembly lines are particularly well adapted to the implementation of the selfbalancing principle. The practical implementations are in expansion, under the simplifying trend of replacing the human operators by robots. From the theoretic perspective, the control is embedded in the system itself. Indeed, from the systemic point of view, the bucket brigades contain both the controlled object, and its controller too; this fact induces a certain difficulty of modelling and analysis. The different approaches belonging to the general theory of dynamic systems have sometimes failed to provide a unitary frame for studying this type of production lines. Although quite difficult, a more refined model of a worker would be of interest. This demands interdisciplinary approaches, taking into account the human aspects, which were neglected by now (motivation, responsibility, psychical state, etc.). From the practical opportunity perspective, the bucket brigades have two major beneficial effects: the production rate increasing and the work-inprocess reduction, both of them due to the fluidization of the productive flux, spontaneously balanced. Because of its simplicity, it is possible that the bucket brigades protocol be easily applicable to a multitude of concrete situations, with the adaptations required in each of cases. ACKNOWLEDGMENTS Antoneta Bratcu was supported by a grant of the Regional Council of Champagne-Ardennes, during the post-doctoral stage that she had at the University of Technology of Troyes, France. REFERENCES Bartholdi, J.J., 0.0. Eisenstein, C. Jacobs-Blecha and H.D. Ratcliff (1995). Design of a bucket brigade production line. (www.isye.gatech.edul-Jjb, May 2002). Bartholdi, J.J. and 0.0. Eisenstein (1996a). The agility of bucket brigade production lines. In: Proceedings of Conference on Flexible and Intelligent Manufacturing, January 22, 1996. Bartholdi, J.J. and 0.0. Eisenstein (1996b). A production line that balances itself. Operations Research, 44/1, 21-34. Bartholdi, J.J., 0.0. Eisenstein and R.D. Foley (200 I). Performance of bucket brigades when work is stochastic. Operations Research, 49/5, 710-719. Bartholdi, U ., 0 .0. Eisenstein and Y.F. Lim (2003). Bucket Brigades on Generalized Assembly

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