Bucket brigades - an example of self-organized production. April 20, 2016

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1 Bucket brigades - an example of self-organized production. April 20, 2016

2 Bucket Brigade Definition A bucket brigade is a flexible worker system defined by rules that determine what each worker does next: Each worker processes his item until he is bumped by a downstream worker, after which he walks back and takes over a job from an upstream worker. Industry examples: Apparel industry Order picking (Amazon)

3 Setup Key assumptions 1 Work content varies continuously along the production line 2 Handover is possible at any position 3 Workers walk back with infinite speed 4 Processing times are deterministic, variation in processing times are due to different worker velocities 5 There is a fixed and well defined ranking of the worker speed. The ranking is constant along the production line.

4 Time evolution of the bucket brigade: Dynamics I Let x m (t) be the position of the worker m at time t and c m his velocity. Assuming no blocking from a downstream worker we have the time evolution of position: x 1 (t) = c 1 t x 2 (t) = c 2 t + x x N (t) = c N t + x N 0 The time to finish the next product is given by x N ( t) = 1. Hence t = 1 x N 0 c N.

5 Dynamics II Poincaré map: P: Σ = {(x 1 ( t), x 2 ( t)...x N 1 ( t), 1)} Σ x 1 n+1 = c 1 c N (1 x N 1 n ) xn+1 2 = c 2 (1 xn N 1 ) + xn 1 c N... xn+1 N 1 = c N 1 (1 xn N 1 ) + xn 1 c N

6 Fixed point Theorem: The Poincaré map has a unique fixed point. Proof: P is an affine map with the linear part A = c 1 c N c 2 c N c N 1 c N det(a Id) is nonzero.

7 Theorem: If workers are ordered along the production line with increasing speed, i.e. c 1 < c 2...c N then the fixed point is globally stable. Proof: Bartholdi and Eisenstein, 1996, For c 1 < c 2...c N blocking is impossible Dynamics is strictly linear with above Poincaré map. Define a j = xj+1 x j c j. Then the Poincaré map becomes a n+1 = T a n Stabiliity I with T = c 0 c c 0 c c N 1 c N 1 c N 1 c N

8 Stabiliity II Proof continued Notice that T is a stochastic matrix with positive entries and whose rows sum to 1. Hence by the following theorems from Markov chains we have asymptotic stability: Theorem: a) The eigenvalues of a stochastic matrix must have modulus less than or equal to 1. b) There is a unique stationary distribution corresponding to eigenvector of T with simple eigenvalue 1.

9 Extensions to inhomogeneous systems. Spatial inhomogeneity Worker skills vary along the production line i.e. no uniform speed ordering. No theory for arbitrary number of workers. Consider two workers, A and B. Scale time and work content such that c B = 1, let worker B start the production line, and { v1 (ξ) for ξ < X c A = v 2 (ξ) for ξ > X with v 1 < 1 < v 2 or v 1 > 1 > v 2.

10 Inhomogeneous systems I Spatial inhomogeneity Case study: v 1, v 2 constant and v 1 < 1 < v 2 q B (t) = t q A (t) = q 0 + v 2 t for q 0 > X q 0 + v 1 t for q 0 < X and t < t x x + v 2 (t t X ) for q 0 < X and t > t X Reset map: Position of worker B when worker A has finished: q n+1 = f (q n ) = { 1 q n v 2 for q n > X X ( 1 v 1 1 v 2 ) qn v v 2 for q n X Most important parameter: f (0) = X v X v 2 and v A = 1/f (0).

11 All possible dynamics for v 1 < 1 when passing is allowed. 1 Examples I x Figure 1: X x, v A > 1: globally stable fixed point x Figure 2: X > x, v A > 1: unstable fixed point, stable period-two orbit

12 Examples II x Figure 3: X x, v A 1: locally stable fixed point, unstable period-two orbit x Figure 4: X > x, v A 1: globally unstable fixed point, reversal of workers

13 General Theory I Lemma 1 There exists a unique fixed point that balances the bucket brigade for all values of v 1 and v 2. Lemma 2 Consider the parameter space (v 1, v 2, X ). The manifold X = 1 1+v 2 separates stable from unstable fixed points. Theorem 1 The two curves f (0) = 1 and X = 1/(1 + v 2 ) determine the average velocity of the second worker and the stability of the fixed point. They intersect in parameter space (v 2, X ) and generate four regions with different dynamical behavior. The resulting dynamics are the ones shown in the figures 1-4

14 General Theory II Theorem 2 All previous theorems are still valid if the velocity of worker A is not piecewise constant but depends on its exact location along the production line as long as there is only one point at which the relative order of the worker s velocities changes. Lemma 3 The throughput for a period-two orbit is always less than the throughput of the fixed point. Lemma 4 Reversing the worker order for a given set of velocities and a given break point interchanges the dynamics between Figure 1 and 4, and between Figure 2 and 3.

15 Conclusion Consequence We may have to choose between self-balancing (e.g. to a limit cycle) and optimal throughput associated with an (unstable) fixed point.

16 Temporal inhomogeneity: Bucket Brigades and Learning Problem Statement One of the basic assumptions of BB: worker speeds are constant in time (though possibly stochastic) Not true in many situations that involve learning and/or forgetting: High turnover production lines - most workers are still on a learning curve Loss and replacement of one experienced worker in an established BB. BB with special skills sections that need constant reinforcement Common to all situations: Learning leeds to nonautonomous dynamical systems

17 Adding a novice to a balanced production line Key assumptions as in standard BB Additionally: n workers arranged from slowest to fastest: c 1 < c 2 <... < c n. one new worker will be added at arbitrary places in the worker ordering. Switching/Passing policy: Workers can pass each other. Handovers will always be done to the nearest worker upstream. Equivalently: Whenever a worker gets blocked he/she will interchange position with the blocking worker.

18 Setup Learning I Starting velocity for new worker: v l < c i i, constant. Potential maximal velocity for new worker: v h. Worker learns only on those parts of the production line where he/she has worked. Learning Rule Wright s model v n = v l tn m t n = Σ n i=0t i The t i are the times between successive finishes of the BB. Towell & Bevis model: v n = v l + (v h v l )(1 e tn/τ ) t n = Σ n i=0t i

19 Learning II Note Wright s model in priciple will allow for unlimited growth but finite resolution will lead to a limiting velocity v h. Production line partitioned into segments of length 0.05.

20 Learning III Examples: final new worker speed profile τ = 0.7, 4 workers + one new worker Starting speed distribution: (c 1, c 2, c 3, c 4 ) = (1.3, 1.6, 2.0, 2.3) Starting speed for new worker v l = 1 Limiting speed for new worker v h = 1.8 Plot of Worker Speeds (1) 2.5 Worker 1 Worker 2 Worker 3 Worker 4 Worker 5 2 Speed Location Figure 1: new worker starts in position 1

21 Examples II Plot of Worker Speeds (2) 2.5 Worker 1 Worker 2 Worker 3 Worker 4 Worker 5 2 Speed Location Figure 2: new worker starts in position 2 Plot of Worker Speeds (3) 2.5 Worker 1 Worker 2 Worker 3 Worker 4 Worker 5 2 Speed Location Figure 3: new worker starts in position 3

22 Examples III Plot of Worker Speeds (4) 2.5 Worker 1 Worker 2 Worker 3 Worker 4 Worker 5 2 Speed Location Figure 4: new worker starts in position 4 Plot of Worker Speeds (5) 2.5 Worker 1 Worker 2 Worker 3 Worker 4 Worker 5 2 Speed Location Figure 5: new worker starts in position 5

23 Result of simulations System stabilizes with new worker in position 1, 2, 3, and 4. Eventual throughput is the same for all worker orders.

24 Analysis I Theorem: The bucket brigade will self organize to the same throughput, independent of initial conditions (initial worker ordering).

25 Analysis II Proof Learning and switching always leeds to an interval of maximal speed I max (obvious) Boundaries of I max are the handover points for the new worker, corresponding to a fixed point of the BB. No periodic orbit inside I max since the motion of the new worker inside I max is completely linear. balanced BB implies that throughput before, after and on I max are equal: v h X Y = Σc i X = Σc j Y

26 Analysis III Proof continued This leads to X = Y = Σc i Σc i + Σc j + v h Σc i + v h Σc i + Σc j + v h which leads to an overall throughput of TP = Σc i + Σc j + v h independent of the position of the new worker.

27 Stability Self Organization of the BB Consider a velocity distribution for the new worker: v l 0 < x < X v(x) = v h X < x < Y v l Y < x < 1 Approximation: Replace the m workers before the new worker and the k workers after the new worker by one worker each with a mean velocity v 1 = Σm i=1 c i m v 2 = Σk j=1 c j k

28 Stability II 2-d Maps Analyze a three worker system that has a fixed point at the starting points. (x 1, x new, x 2 ) = (0, X, Y ). Resulting 2-d map is piecewise linear with four different regions all meeting at the fixed point. Stability of this point is extremely difficult. However: Finite partition into learning region implies X < x new < x 2 < Y.

29 Stability III Implications: for a stability analysis, the new worker always moves with speed v h. Analyze stability of the fixed point of a 3 worker BB with homogeneous velocities v 1, v h, v 2. Necessary and sufficient for stability of the three worker BB is that v 2 > v 1 v 2 > v h v 1

30 Stability IV v Stability region for a 3 worker system v Notice v h might be higher than v 2. Case v) in Example shows v 1 = 1.8, v h = 1.8. Since v new only assymptotically reaches v h we get instability. For stable v h we numerically find that very few transients lead to order changes. For unstable v h order changes are generated and typically lead to a final order from slowest to fastest.

31 1 Plot of Takeovers Chaotic transients Location Time Chaotic transients when worker is placed at unstable position: Transition of new worker from position 3 to position 4.

32 Conclusions Study add a new worker to a stable bucket brigade - new worker learns. Results: BB are very robust and typically adjust to self organized optimal behavior. Positioning of the new worker is irrelevant if BB has a switching rule.

33 Open questions Extend spatial inhomogeneity to n - workers Open questions: Is chaos possible what is the period of the typical orbit for a 3 worker chain? period 3 or period it seems that ordering according to average velocity will lead to locally stable solutions (fixed points or periodic orbits). Is that true for n- workers too? are there practically useful rules for placing workers in an inhomogeneous production line?

Some Generalizations of Bucket Brigade Assembly Lines

Some Generalizations of Bucket Brigade Assembly Lines A Thesis Presented to The Academic Faculty by Yun Fong Lim In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in Industrial