Chaos In Bucket Brigades By Jacob Shrum & Jake Marohl

MAT 452

Traditional assembly lines are rigid systems wherein each worker operates only one

station at which they do one set of tasks before passing the product along to the next worker. The

time it takes to complete a product is simply the time it takes to complete the slowest task in the

line. Therefore, these production systems can be described as a sort of chain, where each link

(worker) is dependent on the next. Moreover, the chain is only as good as its weakest link (the

slowest part of the process). And, perhaps most important of all, should any link in the chain fail

(ie: a worker is injured), the entire production line must stop and wait for the link to be fixed.

Thus it's no surprise that, like the chain which has been replaced by the much stronger and

lighter nylon rope, the traditional production line of the early 1900’s is being brought into the

21st century with a simple, elegant solution known as bucket brigades. While there are many

different forms of bucket brigades, they all share one common feature: There are more machines

than workers, and each worker moves from station to station while following a simple set of

rules that determine when they hand off their work to their successor. The first iteration of such a

production line was known as the “Toyota Sewn Products Management System” or TSS.

The TSS model operates using the following rules (Bartholdi and Eisenstein, 1996 [1]) : ● Number each worker from 1 to n

● Worker i recognizes any worker whose number is less than i as their predecessor

and any worker whose number is greater than i as their successor.

● A station processes at most one item at a time and only one worker is needed

● Each product is identical to the last and all are complete using the same number of

stations

● Each worker processes their item by going from station to station and passes their

item off to another worker using the following rules:

○ Forward Rule: Carry and process your product at each station until you

have either reached the end of the production line - at which point you

follow the backward rule - or until one of your successors takes your item

from you, after which, you follow the backward rule.

○ Backward Rule: Walk backward until you either come upon one of your

predecessors - at which point you will take over their item and begin to

follow the forward rule - or until you reach the first station, at which you

will start a new item and follow the forward rule.

This system is by no means perfect, but many issues that it has can be remedied with simple yet

effective changes to the traditional assembly line model. For example, in order to reduce the amount of

time workers follow the backward rule (wherein they are not working toward the completion of the

product) production lines are often shaped like a U . Thus workers have to walk only a few feet to get to

their predecessor, as opposed to walking down a straight line. Another instance of inefficiency occurs

when workers are faster than their successors, at which point they become blocked. In this situation,

where worker i is blocked by worker i+1, worker i must wait for worker i + 1 to move onto the next

station before they can continue working on their product (because two workers can’t use the same

machine at the same time). In order to get around this, workers are often ordered by speed. One can also

simply reduce the number of workers which, as we will see later, can actually increase the production

rate, but in large systems with many stations, enough workers are required such that ordering them

slowest to fastest is much easier and not affected by events such as worker injury. For the sake of

simplicity, workers will first be modeled with an infinite backward velocity, thus ordering them is a

simple task of arranging them by their forward velocities only. This is reasonable since observations of

TSS lines show no more than a few seconds of backward movement for each worker. This leads us

directly to our first few theorems about this model (Bartholdi and Eisenstein, 1996 [1]) :

Theorem 1: For any TSS line, there exists a fixed point x* = f(x*) That is, there exists a worker

position such that if the worker starts at that position, they will always reset to that position.

Theorem 2: For any TSS line, if the workers are sequenced from slowest to fastest, then any

orbit of worker positions {x(t)=f’(x(0))} converges to a unique fixed point.

Theorem 3: If worker velocities are constant, ordered, and workers are never blocked, then the

TSS line converges exponentially fast to a unique fixed point at which:

1. Workers repeatedly execute the same interval of work content

2. The production rate is the largest possible

These theorems are quite intuitive and well illustrated by Figure 1:

Figure 1 shows 3 workers with velocities v 1 = 2, v2 = 4, and v 3 = 6, with an initial starting

positions of zero (the first station) simulated

for 1000 seconds. For the first 50 seconds,

their paths are quite hectic, but, as theorem

3 suggests, the “paths” very quickly

converge to a unique fixed point, wherein

they all repeatedly execute the same tasks.

In this example, worker 1 returns to position

0, worker 2 to position 18, and worker 3 to

position 54. An evaluation of the local

lyapunov exponents (with 5

partitions) confirms what is

already quite obvious - this

system is stable. The total

production count for this

system is 55 and production is piecewise

linear after 53 seconds. Here, the Largest

Lyapunov Exponent (LLE) is 0.

Blocking: Blocking results from a system

where workers are not ordered from slowest

to fastest. If one considers workers of equal

velocities as in Figure 1 but in a different

order, such as worker 1 third, worker 2 first,

and worker 3 second, the results are drastically different, as shown if Figure 2.

This system, even though the same velocities were used, only produces 50 items in

the same amount of time (a 9.09% reduction in output). This time around we use

more partitions in our calculation of the local lyapunov exponents since the saw

graph indicates that there is no clear formation of sustained stable periodic motion

(The number of partitions is directly related to the compute time, thus it is best

keep them small when we can to save time - of course, one must be careful and

consult the saw graph first). The LLE is 0.044 which is very large for these

systems. One can safely say, by the progression of the local lyapunov exponents,

this system is unstable and the fixed point is a repeller, thus it converges to a limit

cycle with suboptimal production rates. This assertion is consistent with those of Bartholdi et al.

Quasi-Periodic Motion: As Bartholdi et al. point out, the

system with 3 workers of velocities (2,1,2) is quasi

periodic. Noting that the workers’ behavior is

“predictable, but not periodic” and that “all orbits

converge to the periphery of an ellipse”. While this

certainly isn’t the most substantial piece of the paper, it

deserves more focus than it gets. Thus we shall back up

Bartholdi et al’s claims. Looking at the saw tooth graph in

Figure 4 we see, specifically by looking at worker 2’s

position, that there seems to be some level of periodicity

as the ‘cycle times’ between local maxima are rather

consistent - but not equal. Moreover, there is some hint of

a pattern of 3 local minima (0) between each successive

local maxima of worker 1. Because this system seems

rather chaotic, 250 partitions were used over the 1000

iteration time series to calculate the local lyapunov

exponents. In Figure 5 we can see that the system is more

often unstable than it is stable - though dips into negative

LE are less common and short lived, they are much greater

in magnitude. This figure also allows us to confirm that

there are indeed chaotic orbits (We will see later that not

all systems of this type are unstable. In fact, most we

found were stable). Finally, in the next figure, we verify that the orbits converge to the periphery of an

ellipse.

In Figure 6 we see worker 1 plotted against worker 2 as well as a fixed point line. This shows the edge

of the ellipse. In Figure 7, all 3 workers are plotted against each other, revealing the ellipse. This shape

appears whenever the first and last workers have the

same speed and are both faster than the second worker -

but not all arrangements of this type are quasi-periodic;

some are simply periodic. For example, the system

(10,2,10) quickly converges to a periodic orbit and thus

has a series of unique fixed points which workers

repeatedly reset to. This is evident from the sawtooth

graph in Figure 8 and the LLE graph in Figure 9..

Order Over Speed: By theorem 3, we know that having ordered workers is much more important than

having fast workers. For example the system of workers (2,10,1) completes 33 items in 1000 seconds,

while the system (1,2,3) completes 55 items in the same amount of time. Along the same lines, when

adding a worker, preserving order is crucial to the production rate of the assembly line. Consider a

rearrangement of the optimal system (1,2,3)

such that worker 1 swaps location with worker

2, and worker 3 sits on the sidelines. Therefore

the system can be thought of as (2,3,null). This

is shown in Figures 10 and 11. For better

visibility, this simulation was run for 9999

iterations. For the first third, the system

(2,3,null) is active, then worker 3 is added with

a velocity of 1 making the system (2,3,1) and

as we can see, the production rate goes down. Finally, the system is reordered to (1,2,3) to

illustrate that adding this worker can be

advantageous but only if done correctly. This

simulation also provides a great example of

theorem two. In the beginning, the workers are

ordered and thus their orbits converge to

unique fixed point. Then, as worker 3 is added

and proceeds to block worker 2 and worker 1,

the system diverges to that of one where

worker 1 (as well worker 2) simply go up and

down the line. Once the workers are reordered,

their orbits converge to a fixed point rapidly.

This suggests that, should a bucket brigade

become unordered, restoring order - by

rearranging workers or simply removing them -

will allow the assembly line to regain optimal

production rates.

A Model With Overtaking: Here we focus on the model described by Bartholdi et al. in Deterministic

Chaos in a Model of Discrete Manufacturing. The rules are almost entirely the same as TSS, but the few

changes make a world of difference. In this model, workers can pass each other and have a constant

non-infinite backwards velocity.

Unlike the first model, there is no clear position function or way to iterate it. Instead, the model

needs to be analysed using the hand-off points, or the location at which the workers hand off items.

Notice that, with this model, the hand-offs do not always occur at the same time. It is even possible there

is no hand-off point, such as when two workers start at the origin with velocities v 1=3, w 1=1, v2=1,

w 2=3.

As in the model with an infinite backwards velocity and without passing, the workers can be arranged so

that the system is stable. The order in which they must be arranged is given by the Convergence Condition,

shown in Equation 1 (Bartholdi, Eisenstein and Lim, 2009 [2]) . Equation 1:

The Convergence Condition.

..1v1

− 1w1

> 1v2

− 1w2

> . 1vn

− 1wn

When workers are arranged according to the Convergence Condition, the fixed positions of the hand-off

points will be attractors. This means that each worker will converge to a section of the assembly line where they

repeat the same interval of work without passing other workers.

This model has three theorems (Bartholdi, Eisenstein and Lim, 2009 [2]) , which are as follows:

Theorem 4: For any bucket brigade, the point for i=0, 1, … n is a fixed pointxi* =

+ ∑n

1( 1

vi

1w

i) −1

+ ∑i

1( 1

vi

1w

i) −1

on the map that relates successive points of hand-offs and is a unique point of balance. A point of

balance is unique when the work is partitioned among the workers.

Theorem 5: If workers are arranged in accordance to the convergence condition, then x* is an

attractor.

Theorem 6: There exists a two-worker bucket brigade in which the sequence of hand-off points

is chaotic.

These theorems are clearly shown in the following figures.

Figure 12 shows a the hand-off locations as time passes of a stable system with workers v 1=1, w 1=.5,

v2=1, w 2=.25, v3=1, w 3=.125 at initial positions (0,20,50).

The hand-off points rapidly

approach a fixed point, which means

that the fixed point is an attractor, as

stated in Theorem 5. The hand-off

points are x 1* = 51.75 and x 2* =

82.75, which supports Theorem 4,

which calculates the points as x 1* =

51.72 and x 2* = 82.76, The error in

the experimental value of x 1* is

most likely caused by code that

determines when a worker has

reached the beginning or end of the

line. In the program, workers move

forward or backward and their positions are checked to ensure one has not walked past either end of the

line; If one has, its position is set such that it now stands at the end and waits till the next second to

change direction. Small errors can propagate in this way because the true position of the worker would

not be the end of the line, but would instead be

slightly inside it. A smaller step size would

decrease this error.

Figure 13 shows the positions of the

workers as time progresses. As is demonstrated

after t = 700, the final hand-off points show a

point of balance, with each worker repeatedly

performing a unique subsection of work.

Figures 14 and 15 show a chaotic system

with workers v 1=1, w 1=1/3, v2=1, w 2=1, v3=2/3,

w 3=4/3at initial positions (0,20,50). Unlike in

Figure 12, Figure 14 does not converge to a set of fixed hand-off points. Instead, its hand-off points

jump around chaotically. Figure 15 shows that the workers are not performing unique sections of work,

and therefore are not on a point of balance. The products are started and finished randomly, shown when

workers reverse directions at x=0 and x=1. This would lead to problems when other processes are

implemented before or after the assembly line.

Resiliency of Stable Systems: Because each system is a model of real-world behavior, it is unlikely that

the exact initial conditions are known. Any deviation in the initial conditions has the potential to throw

an otherwise balanced system into chaos. To evaluate the sensitivity of these systems, each of the stable

models for both styles of bucket brigade were modified in the following way:

1. Before each simulation, define two random variables with uniform distribution r, s such that they

both have a range of (0,1).

2. If s >= 0.5, r = -r

3. V 1 is modified such that . This allows for the very likely possibility that a1 ) V 1 = V 1 * ( + r

10

worker won't be able to maintain a perfectly constant speed from day to day. For each simulation

there is deviation from the worker’s ideal velocity of up to 10%.

4. For both the forward and backward velocity of each worker i, repeat steps 1 - 3

5. Start the simulation. Workers velocities are not modified while the simulation is running (it was

found to have very little effect up until the max deviation exceeded 100%, which is

unreasonable, especially from one moment to the next.

The program that has been used to simulate workers in bucket brigade was modified to display 1 graph

of the ideal system (this serves as a control) and 7 graphs of simulations with worker velocities that

randomly deviate by up to |10%| from those in the control. It then repeats this process 1000 times for a

total of 8000 simulations, 7000 of which are not the control. In figure 16 we consider the convergent

system: v 1=1, w 1=1, v2=1, w 2=.5, v3=1, w 3=.25

On the top left is a graph of the hand-off points of the control (unmodified system). Each of the

remaining seven graphs represent a randomly modified system. On the bottom left is a graph of a system

that we consider to have failed to converge to a fixed point within the time series (2.77 hours), and thus

fails to achieve the optimal output, given the target velocities, within any reasonable amount of time.

After looking over the hand-off point graphs of over 12000 modified systems, only 5 were found to not

converge by the end of the time series or were chaotic. For this system, a variation in velocity ranging

from -10% to 10% for each worker leads to non optimal output caused by chaos or very slow

convergence in only 0.04% of all cases. For a TSS model with v 1 = 1, v2 = .5, and v 3 = .25, after 21,000

trials at 10% maximum deviation from the control, we concluded that the TSS system is very resilient as

there were no sightings of chaos or failure to converge to a fixed point. In fact, no such thing occurred

until we allowed for a maximum deviation of 20%, at which point the line broke down catastrophically

1% of the time.

Conclusion: In this paper, we experimentally examined the properties of bucket brigades in two

different models. We showed the initial conditions in which each model is stable or unstable, and we

demonstrated their behaviors. We examined the initial conditions that result in the system converging to

an ellipse, as mentioned is possible by Bartholdi et al. We also experimentally analyzed the difference of

production output between an ordered system of workers and an unordered but larger and faster system

of workers. Finally, we investigated the possibility a system will become unstable when the velocities of

the workers vary or are not accurately known.

References

[1] Bartholdi, J. and Eisenstein, D. (1996). A Production Line that Balances Itself. Operations Research , 44(1), pp.21-34.

[2] Bartholdi, J., Eisenstein, D. and Lim, Y. (2009). Deterministic chaos in a model of discrete

manufacturing. Naval Research Logistics , 56(4), pp.293-299.