Bucket brigades - an example of self-organized
production.
April 20, 2016
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)
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.
Dynamics I
Time evolution of the bucket brigade:
Let x m (t) be the position of the worker m at time t and cm his
velocity.
Assuming no blocking from a downstream worker we have the time
evolution of position:
x 1 (t) = c1 t
x 2 (t) = c2 t + x02
.....
N
x (t) = cN t + x0N
The time to finish the next product is given by x N (t̄) = 1. Hence
t̄ =
1 − x0N
.
cN
Dynamics II
Poincaré map:
P: Σ = {(x 1 (t̄), x 2 (t̄)...x N−1 (t̄), 1)} → Σ
1
xn+1
=
2
=
xn+1
....
N−1
xn+1
=
c1
(1 − xnN−1 )
cN
c2
(1 − xnN−1 ) + xn1
cN
cN−1
(1 − xnN−1 ) + xn1
cN
Fixed point
Theorem:
The Poincaré map has a unique fixed point.
Proof:
P is an affine map with the linear part

0 0... − cc1
 1 0... − cN2

cN
A=
 ......
c
0... 1 − N−1
cN
det(A − Id) is nonzero.





Stabiliity I
Theorem:
If workers are ordered along the production line with increasing
speed, i.e. c1 < c2 ...cN then the fixed point is globally stable.
Proof: Bartholdi and Eisenstein, 1996,
• For c1 < c2 ...cN blocking is impossible
• Dynamics is strictly linear with above Poincaré map.
j+1
j
• Define aj = x c−x . Then the Poincaré map becomes
j
an+1 = T an
with

0
 cc0
1
T =
 ......
0...
0...
0...
cN−1
cN
1
1 − cc10
1−
cN−1
cN




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.
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 cB = 1, let worker B start
the production line, and
v1 (ξ) for ξ < X
cA =
v2 (ξ) for ξ > X
with v1 < 1 < v2 or v1 > 1 > v2 .
Inhomogeneous systems I
Spatial inhomogeneity
Case study: v1 , v2 constant and v1 < 1 < v2
qB (t) = t
 0
 q + v2 t
q 0 + v1 t
qA (t) =

x + v2 (t − tX )
for q 0 > X
for q 0 < X and t < tx
for q 0 < X and t > tX
Reset map: Position of worker B when worker A has finished:
q n+1 = f (q n )
(
=
1−q n
v2
X ( v11
−
qn
v1
+
1
v2
Most important parameter: f (0) =
X
v1
+
1−X
v2
−
1
v2 )
for q n > X
for q n ≤ X
and v̄A = 1/f (0).
Examples I
All possible dynamics for v1 < 1 when passing is allowed.
1
0.8
0.6
0.4
0.2
0
0.2
0.4
x
0.6
Figure 1: X ≤
x ∗,
1
0.8
1
v̄A > 1: globally stable fixed point
0.8
0.6
0.4
0.2
0
0.2
0.4
x
0.6
Figure 2: X >
orbit
x ∗,
0.8
1
v̄A > 1: unstable fixed point, stable period-two
Examples II
1
0.8
0.6
0.4
0.2
0
0.2
0.4
x
0.6
x ∗,
Figure 3: X ≤
period-two orbit
0.8
1
v̄A ≤ 1: locally stable fixed point, unstable
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.2
0.4
x
0.6
Figure 4: X >
of workers
x ∗,
0.8
1
v̄A ≤ 1: globally unstable fixed point, reversal
General Theory I
Lemma 1
There exists a unique fixed point that balances the bucket brigade
for all values of v1 and v2 .
Lemma 2
Consider the parameter space (v1 , v2 , X ). The manifold X =
separates stable from unstable fixed points.
1
1+v2
Theorem 1
The two curves f (0) = 1 and X = 1/(1 + v2 ) determine the
average velocity of the second worker and the stability of the fixed
point. They intersect in parameter space (v2 , X ) and generate four
regions with different dynamical behavior. The resulting dynamics
are the ones shown in the figures 1- 4
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.
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.
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
Adding a novice to a balanced production line
Key assumptions as in standard BB
Additionally:
• n workers arranged from slowest to fastest: c1 < c2 < ... < cn .
• 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.
Learning I
Setup
Starting velocity for new worker: v` < ci ∀i, constant. Potential
maximal velocity for new worker: vh . Worker learns only on those
parts of the production line where he/she has worked.
Learning Rule
• Wright’s model
vn = v` tnm
tn = Σni=0 ti
The ti are the times between successive finishes of the BB.
• Towell & Bevis model:
vn = v` + (vh − v` )(1 − e −tn /τ )
tn = Σni=0 ti
Learning II
Note
• Wright’s model in priciple will allow for unlimited growth but
finite resolution will lead to a limiting velocity vh .
• Production line partitioned into segments of length 0.05.
Learning III
Examples: final new worker speed profile
τ = 0.7, 4 workers + one new worker
Starting speed distribution: (c1 , c2 , c3 , c4 ) = (1.3, 1.6, 2.0, 2.3)
Starting speed for new worker v` = 1
Limiting speed for new worker vh = 1.8
Plot of Worker Speeds (1)
Worker
Worker
Worker
Worker
Worker
2.5
1
2
3
4
5
Speed
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Location
Figure 1: new worker starts in position 1
0.8
0.9
1
Examples II
Plot of Worker Speeds (2)
Worker
Worker
Worker
Worker
Worker
2.5
1
2
3
4
5
Speed
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Location
Figure 2: new worker starts in position 2
Plot of Worker Speeds (3)
Worker 1
Worker 2
Worker 3
Worker 4
Worker 5
2.5
Speed
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Location
Figure 3: new worker starts in position 3
Examples III
Plot of Worker Speeds (4)
Worker
Worker
Worker
Worker
Worker
2.5
1
2
3
4
5
Speed
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Location
Figure 4: new worker starts in position 4
Plot of Worker Speeds (5)
Worker 1
Worker 2
Worker 3
Worker 4
Worker 5
2.5
Speed
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Location
Figure 5: new worker starts in position 5
Result of simulations
System stabilizes with new worker in position 1, 2, 3, and 4.
Eventual throughput is the same for all worker orders.
Analysis I
Theorem:
The bucket brigade will self organize to the same throughput,
independent of initial conditions (initial worker ordering).
Analysis II
Proof
• Learning and switching always leeds to an interval of maximal
speed Imax (obvious)
• Boundaries of Imax are the handover points for the new
worker, corresponding to a fixed point of the BB. No periodic
orbit inside Imax since the motion of the new worker inside
Imax is completely linear.
• balanced BB implies that throughput before, after and on Imax
are equal:
Σcj
vh
Σci
=
=
X −Y
X
Y
Analysis III
Proof continued
This leads to
X
=
Y
=
Σci
Σci + Σcj + vh
Σci + vh
Σci + Σcj + vh
which leads to an overall throughput of
TP = Σci + Σcj + vh
independent of the position of the new worker.
Stability
Self Organization of the BB
Consider a velocity distribution for the new worker:

 v` 0 < x < X
v
X <x <Y
v (x) =
 h
v` 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
v1 =
v2 =
Σm
i=1 ci
m
Σkj=1 cj
k
Stability II
2-d Maps
Analyze a three worker system that has a fixed point at the
starting points .
∗
(x1∗ , xnew
, x2∗ ) = (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 < xnew
< x2∗ < Y .
Stability III
Implications:
• for a stability analysis, the new worker always moves with
speed vh .
• Analyze stability of the fixed point of a 3 worker BB with
homogeneous velocities v1 , vh , v2 .
Necessary and sufficient for stability of the three worker BB is that
v2 > v1
v2 > vh − v1
Stability IV
1
0.8
0.6
v2
0.4
0.2
0
0
0.2
Stability region for a 3 worker system.
0.4
0.6
0.8
1
v1
Notice
• vh might be higher than v2 .
• Case v) in Example shows v1 = 1.8, vh = 1.8. Since vnew only
assymptotically reaches vh we get instability.
• For stable vh we numerically find that very few transients lead
to order changes.
• For unstable vh order changes are generated and typically lead
to a final order from slowest to fastest.
Chaotic transients
Plot of Takeovers
1
0.9
0.8
0.7
Location
0.6
0.5
0.4
0.3
0.2
0.1
0
0
50
100
150
Time
200
250
300
Chaotic transients when worker is placed at unstable position:
Transition of new worker from position 3 to position 4.
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.
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 23−1 .
• 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?