The Effects of Worker Learning, Forgetting,
and Heterogeneity on Assembly Line
Productivity
Scott M. Shafer • David A. Nembhard • Mustafa V. Uzumeri
Babcock Graduate School of Management, Wake Forest University, P.O. Box 7659,
Winston-Salem, North Carolina 27109-7659
Department of Industrial Engineering, University of Wisconsin-Madison, 1513 University Avenue,
Madison, Wisconsin 53706-1572
Department of Management, Auburn University, 415 W. Magnolia Avenue, Auburn, Alabama 36849-5241
[email protected] • [email protected] • [email protected]
T
he authors investigate through several simulations how patterns of learning and forgetting affect the operating performance of an assembly line. A unique aspect of this
study is that a distribution of learning/forgetting behavior based on an empirical population of workers is used rather than assuming the same learning pattern for all employees.
The paper demonstrates that modeling only central tendency and not the variations across
workers tends to systematically underestimate overall productivity. The data used to estimate
the parameters for the distribution of learning curves were collected from an assembly line
that produces car radios.
Analysis of the models fit to a population of workers reveals that higher levels of previous experience are positively correlated with higher steady-state productivity levels and
negatively correlated with the learning rate. To further motivate the study, a conceptual
model with several factors hypothesized to influence assembly line productivity is presented.
Among key factors included in the model are the rate of worker learning, the size of the
worker pool, task tenure, and the magnitude of worker forgetting. In controlled computer
simulation experiments, each of these factors was found to be statistically significant, as were
a number of the two-way interaction terms.
(Learning; Forgetting; Worker Heterogeneity; Simulation)
1.
Introduction
environment, the individual worker must constantly
adapt to changing tasks.
Furthermore, as the frequency with which individual workers are required to master new tasks
increases, the amount of time the workforce spends
on the steep part of the learning curve increases as
new products are introduced, as they are rotated to
other jobs, and as their work is restructured (e.g.,
Uzumeri and Nembhard 1998, Uzumeri and Sanderson 1995, Brown and Duguid 1991). More specifically,
in less dynamic environments workers have ample
0025-1909/01/4712/1639$5.00
1526-5501 electronic ISSN
Management Science © 2001 INFORMS
Vol. 47, No. 12, December 2001 pp. 1639–1653
A variety of forces are simultaneously creating a new
learning environment for shop floor workers. For
example, many organizations are shifting from making a few products with long life cycles to managing product families with shrinking life cycles.
At the same time, many of these organizations are
accelerating their rate of process innovation and
improvement, increasing worker flexibility through
cross-training, and restructuring or reorganizing work
activities. An important implication is that in this new
SHAFER, NEMBHARD, AND UZUMERI
Worker Learning, Forgetting, and Heterogeneity
opportunity to progress up the learning curve and
operate at a highly proficient rate for an extended
period of time. This is in stark contrast to the more
dynamic environments characteristic of the present.
In these environments, workers are often not afforded
sufficient time to achieve a high level of proficiency
at a particular task before reassignment. However,
workers in this environment are observed “learning
to learn” with increased experience switching tasks.
For example, Nembhard (2000) observed higher individual learning rates and higher individual forgetting
rates for workers with higher experience levels on an
inspection task similar to that in the current study.
Nembhard and Uzumeri (2000a) observed a similar
pattern of higher learning and forgetting rates for
both manual and cognitive tasks in industry.
From the organization’s point of view, key concerns
arise from the potential consequences due to lost
output, higher cost, and the general competitiveness
of the organization when its workforce spends proportionally more time in the learning process. Once
these consequences are quantified, it is logical to next
consider what options are available to help mitigate
them. For example, can new technologies be deployed
to improve the rate of worker learning and/or reduce
the impact of worker forgetting?
The purpose of this study is to investigate how heterogeneity of worker learning and forgetting affects
the operating performance of an assembly line under
a variety of managerially controllable conditions. A
related general research question naturally arises, one
that is not addressed in the literature: What is the
impact of not modeling the heterogeneity inherent in
real populations of workers, as has been common in
the literature (e.g., McCreery and Krajewski 1999)?
In investigating this issue, we will illustrate that modeling only central tendency and not also variation
among workers can result in substantially underestimating overall system productivity. Proof of this
assertion is given in the appendix. Therefore, in addition to obtaining a closer representation of worker
learning and forgetting patterns from an actual process, we also address the need for empirical investigation of learning and forgetting heterogeneity, as
called for by several researchers (e.g., McCreery and
Krajewski 1999, Lance et al. 1998).
1640
Regarding the question of how patterns of worker
learning and forgetting affect the operating performance of an assembly line, our results demonstrate
that increasing the rate of learning, perhaps via the
adoption of new training technologies, can moderate the negative impacts of both increases in the size
of the worker pool and worker forgetting. Furthermore, while overall system performance generally
deteriorated as the size of the worker pool increased,
longer task tenures did not automatically translate
into increased system performance.
2.
Overview of Previous Research
The literature relevant to this study is broad, touching on many topic areas. For example, Yelle (1979)
and Belkaoui (1989) provide surveys of the learning
curve literature. Nembhard and Uzumeri (2000a) provide empirical comparisons of many common learning curve models. However, forgetting has received
relatively little empirical attention in the literature. In
overviewing research related to organizational forgetting, Argote (1996) notes that knowledge can reside in
the organization’s employees, its technology, and in
its structure. Accordingly, the level of knowledge can
depreciate due to personnel turnover and when technology is not accessible or easily used. Argote (1996)
also discusses the impact of turnover on the productivity of production workers. The present study
extends this research by investigating how the size
of the worker pool and the length of task tenures
impact worker forgetting and also the extent to which
the rate of learning can, in the long run, mitigate the
effects of worker forgetting.
Models of worker performance in the literature
have relatively infrequently considered the learning and forgetting phenomena. There have been
recent efforts to incorporate these effects into performance models because industry is recognizing the
losses they incur due to relearning (e.g., Kher 2000,
McCreery and Krajewski 1999). We remark that it is
common in such studies to employ a homogeneous
workforce with respect to learning and forgetting.
This has been at least in part due to, as stated by
McCreery and Krajewski (1999, p. 2034) the fact that
none of the literature they reviewed “provided an
Management Science/Vol. 47, No. 12, December 2001
SHAFER, NEMBHARD, AND UZUMERI
Worker Learning, Forgetting, and Heterogeneity
Table 1
Individual Work History Characteristics of Empirical Data
No. of units produced
Total no. of days worked (excluding breaks)
Production time per unit (minutes)
Task tenure (days)
No. of breaks∗
Break length (days)
∗
M
SD
Max
2389
252
320
560
39
1143
1496
163
219
494
22
891
7719
691
1482
1280
100
2151
2266
235
237
436
30
960
Min
364
30
103
172
00
483
Based on gaps in production of 50 hours or more (i.e., longer than one weekend).
empirically derived function of the forgetting phenomenon.” Studies by Nembhard (2000) and Nembhard and Uzumeri (2000a) illustrate that empirical
distributions of worker learning and forgetting are
valuable in showing the variation present in workers
performing real on-the-job tasks. It is an important
contribution of the current study that a heterogeneous
workforce is modeled and that the nature of the heterogeneity is informed by empirical data, thus tying
our simulation study closely to an operating production process.
In this paper we simulate a set of assembly line
test stations to examine the effect that spreading task
experience across a larger pool of workers has on
system productivity considering learning and forgetting effects. Second, we investigate the impact of task
tenure in an environment where individuals work
independently of one another. Third, we examine the
relative effects that changes in the learning and forgetting rates have on overall productivity. Furthermore,
these factors are investigated in an environment with
a heterogeneous workforce.
3.
Mdn
Methodology
3.1. Data Collection
To investigate how heterogeneous learning and forgetting affects assembly line performance, data were
collected from the final test and inspection station of
an assembly line that produces car radios. The line
began production ramp-up in late April 1996. Over
the summer of 1996 additional lines and personnel
were added, culminating in August 1996 with a final
configuration of 24 test stations on three separate lines
(or eight test stations per line). Inspection time data
Management Science/Vol. 47, No. 12, December 2001
were collected for 176,000 items produced by 75 workers1 through 1996. Of the 176,000 radios inspected,
only 350 failed the final testing.
The final assembly test process is a combination of machine-paced and human-paced activities.
The machine-paced activities are computer controlled
and thus exhibit minimum variability. The workers
also perform a number of supplementary functional
and cosmetic inspections according to a documented
ISO 9000 inspection procedure. In total, the inspection
procedure requires 130 distinct evaluation criteria and
operational steps. The final assembly test workers are
trained using the written procedures and through onthe-job training.
A typical cycle begins when the inspector picks
a radio from the adjacent conveyor and begins a
series of cosmetic tests. After completing these manual tasks, the unit is placed in a fixture where a
computer tests the internal electronic functions. The
inspector then either places a shipping label on the
units that pass or identifies the failure and routes
it to rework. At the time of the study, the inspection stations were the bottleneck operation for the
entire assembly line. Thus, the inspection station performance provided a good proxy for the performance
of the entire line. Some descriptive characteristics of
these data are given in Table 1, where the size of the
1
The study period data involved a total of 148 workers, 75 of whom
performed long enough to obtain reasonable fits to the model. The
73 workers that were excluded from the study each worked less
than one day and produced relatively few total units. Estimates of
long-run productivity would be relatively unreliable based on one
day’s practice. Since these workers also produced at relatively low
production rates, being on the steepest portion of their learning
curves, our results may be considered to be relatively conservative.
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SHAFER, NEMBHARD, AND UZUMERI
Worker Learning, Forgetting, and Heterogeneity
worker pool was constant, and there were negligible learning effects present at the aggregate organizational level. We remark that in aggregate, employees worked and took breaks from the inspection task
more or less at random due to various cross-training
programs, seniority-based bumping, sick leaves, and
vacations that occurred during the study period. As a
result, the task tenure and the number and lengths of
the breaks varied among workers.
3.2. Modeling Worker Learning and Forgetting
Each worker’s output history is recorded in buckets of
20 consecutive radios, where the average production
time for each bucket corresponds to approximately
30 minutes of production for an average worker.2
To obtain best-fit parameter estimates we follow a
framework proposed by Uzumeri and Nembhard
(1998) for fitting curves to each individual’s performance history. In the first phase of this framework,
a common mathematical function is fit to the timeseries performance data for each worker in the population.3 The resulting distribution of best-fit parameter estimates is used to describe the behavior of the
group. The criteria used in selecting a model includes:
1) minimize average of the mean squared errors
(MSEs) from the individual fitted models, 2) minimize
the standard deviation of the MSEs, 3) minimize the
number of model parameters, and 4) give preference
to the models that have clearly interpretable parameter definitions.
In a related study, Nembhard and Uzumeri (2000b)
determined that of 11 common learning curve models
selected from the literature, a three-parameter hyperbolic model suggested by Mazur and Hastie (1978)
2
Over 83% of the total variation between individual units is
removed in the bucketing process, with negligible signal loss.
That is, with unit cycle times as small as 40 seconds, the unpredictable within-bucket variation arises from numerous unmeasured
sources, and at the same time workers are not exhibiting significant improvement within these 30-minute spans. The smoothing
accomplished by the bucketing process allows for more reliable and
consistent convergence in the model-fitting process. As a result, we
obtained convergence for all 75 workers with an average R2 statistic
equal to 96% and standard deviation of 2%.
3
A nonlinear least-squares-curve-fitting procedure (SAS PROC
NLIN) was used.
1642
performed best in terms of Criteria 1 and 2. Nembhard and Uzumeri found that the traditional loglinear model performed reasonably well, but did not
fit the wide range of empirical learners as well as
the hyperbolic model. In that study the R2 statistics
were above .98 in 90% of 3,874 individual learning
episodes. Based on these results, the three-parameter
hyperbolic model was initially selected for the present
study. The R2 statistics for learning episodes in the
current study data averaged over 96%, which we
found comparable to the earlier study. The hyperbolic
model of learning is of the following form:
x+p
yx = k
x+p+r
s.t. y k p x ≥ 0 and p + r > 0
(1)
where yx is a measure of the productivity rate corresponding to x units of cumulative (or total) work.
Fitted parameter p represents the prior expertise
attributable to the task based on a fit of the model
to the data and may be viewed as an estimate of
a workers’ expertise acquired from past and similar
experience. It is, in effect, shifting the learning curve
backward in cumulative work to estimate prior expertise. The fitted parameter k estimates the asymptotic
steady-state productivity rate, which is the rate that
can be expected once all learning has been completed.
The fitted parameter r is the cumulative production
and prior expertise required to reach k/2, starting
from the production rate corresponding to zero cumulative work and prior expertise. Thus, r represents the
learning relative to the individual’s steady-state productivity rate, k. Note that smaller values of r correspond to a more rapid approach to steady-state or
faster learning. This model is capable of describing
both positive and negative learning episodes, as were
observed in the data.
Forgetting of tasks, in practical settings, tends to
occur after intermittent breaks (i.e., gaps or interruptions that occur during particular work assignment)
in production. While management may have the ability to influence the average rate of breaks in production, the actual number, times, and lengths of these
breaks depend on various things outside of direct
managerial control, including illness, vacation, and
Management Science/Vol. 47, No. 12, December 2001
SHAFER, NEMBHARD, AND UZUMERI
Worker Learning, Forgetting, and Heterogeneity
seniority-based bumping. Many forgetting models in
the literature are designed to handle a single break of
a given length, which for intermittent breaks would
require the reapplication of such models for each and
every break (e.g., Globerson et al. 1989, Bailey 1989).
We remark that given substantial quantities of production data, this prospect may be relatively cumbersome to perform. The modifications to Equation
(1), which follow, were introduced by Nembhard and
Uzumeri (2000a) to include the effects of intermittent
forgetting. These modifications were shown to perform well in measuring forgetting in both manual and
cognitive tasks (Nembhard and Uzumeri 2000a).
Forgetting is modeled based on a measure termed
recency of experiential learning, R, which provides a
relative measure of how recently an individual’s practice was obtained. For each unit of a worker’s cumulative (total) production x, we determine the corresponding recency measure, Rx , by computing the ratio
of the average elapsed time to the elapsed time for
the most recent unit produced, as in Equation (2).
The elapsed time for unit x for a particular worker is
given by tx − t0 , which is the difference between the
timestamps of the start of the current unit, tx , and the
earliest timestamp for the worker, t0 .
x
t − t (2)
Rx = i=1 i 0 x
tx − t0 We note that Rx will be bounded below by 0 and
above by 1, in such a manner that values approaching
1 indicate that all experience was obtained immediately preceding the current unit, and values approaching 0 indicate that experience was obtained in the
distant past. For a constant productivity rate, the
recency, Rx , tends toward a nominal value of 0.5. To
incorporate the effect that the recency of experience
has on individuals within the population of workers, the cumulative production x is discounted (or
reduced) by the corresponding factor Rx , where the
fitted parameter represents the degree to which the
individual forgets the task. Restricting > 0, we note
that for small values of there is very little discounting of cumulative work. As increases, the term
Rx becomes smaller, reflecting a greater discounting of the cumulative work x and greater forgetting.
Management Science/Vol. 47, No. 12, December 2001
The model of individual learning and forgetting is
given by
xRx + p
yx = k
xRx + p + r
y k p x ≥ 0 p + r > 0
(3)
Although the parameters in Equation (3) allow
for the description of a family of curves including both positive and negative learning, the focus
of the present study was limited to cases of positive learning. That is, workers whose performance
deteriorated as they gained experience (i.e., exhibited
negative learning) were not included in this study.
While there is evidence of the existence of negative learning (e.g., Nembhard and Uzumeri 2000a), at
present the phenomenon is poorly understood. Thus,
we only included workers with positive learning patterns. Five of the 75 workers who had work histories
of more than one day exhibited negative learning and
were not included in the simulation study.
Table 2 summarizes the results of the learning
curves fit to the 70 workers who performed a nonnegligible amount of work and exhibited positive learning. On average, the steady-state productivity rate
for the radio inspection task after all learning had
occurred was 29.7 radios per hour. The fastest worker
in the sample would be expected to achieve a productivity rate of 45 radios per hour, while the slowest worker in the sample would likely achieve a productivity rate of 19.4 radios per hour. We obtained
convergence for each of the 70 workers with significant parameter estimates (5% significance level) and
an average R2 statistic of 96.6% (standard deviation
2%).
Further inspection of Table 2 suggests that the
workers, on average, had the equivalent experience
of inspecting 1,150.5 radios prior to the current study.
Table 2
M
SD
Maximum
Minimum
Summary of Best-Fit Parameters to Inspection Time Series
Data
k (radios/hr)
p (radios)
r (radios)
297
74
450
194
11505
24961
131657
00
9211
22893
149809
13
1.3
1.8
5.0
0.0
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SHAFER, NEMBHARD, AND UZUMERI
Worker Learning, Forgetting, and Heterogeneity
Table 3
Correlation Coefficients Between Best-Fit Parameters and Work History Characteristics
k
p
r
Units
Produced
No.
Breaks
Total Days
Worked
(Excluding
Breaks)
Task
Tenure
Break
Length
(Days)
∗∗
p
0505
r
0658∗∗ 0827∗∗
0086 −0249∗
0244∗
Units produced
0128
0117
0086
0204
No. breaks
0030
0054
0013
0177 0639∗∗
Total days worked
0030
0025
0014
0111 0849∗∗
0557∗∗
(excluding breaks)
Task tenure
0103
0017
0050 −0022 0281∗ −0258∗
Break length (days)
−0121 −0160 −0137 −0073 −0207
−0225
Average production
−0269∗ −0117
0031 −0159 −0464∗∗ −0337
time per unit (minutes)
∗
0560
−0158
−0338∗∗
0015
−0009
−0127
Pearson correlation is significant at the 0.05 level (2-tailed).
Pearson correlation is significant at the 0.01 level (2-tailed).
∗∗
The worker with the most experience had the equivalent experience of inspecting over 13,000 radios, while
12 of the workers appeared to have no equivalent past
experience. In terms of the rate of learning, on average the workers needed to inspect 921 radios to reach
half their potential steady-state productivity rate. The
slowest learner in the study required the experience
of inspecting almost 15,000 radios to achieve this level
of proficiency, while the fastest learner was able to
achieve this proficiency after inspecting fewer than
two radios. Finally, the average rate of forgetting, ,
was 1.3, with a minimum of 0.0 and a maximum of
5.0. As this discussion illustrates, Equation (3) succinctly describes individual patterns of learning and
forgetting on the basis of four parameters (k, p, r, and
). In effect, these four parameters describe the learning and forgetting patterns recorded in hundreds of
data points for each worker. Extending this, the entire
population of workers can be viewed as the set of
best-fit learning curves. The result is a distribution of
mathematical curves that describes the workforce.
To obtain additional insights into the patterns of
learning and forgetting exhibited by the workers
studied, the pairwise correlation coefficients were
calculated between the four parameters and five
additional work history characteristics. As shown in
Table 3, the correlation coefficients between k, p,
and r were all statistically significant and positive.
1644
For the workers studied, higher levels of previous
experience were correlated with higher steady-state
productivity levels. On the other hand, because the
rate of learning decreases as r increases, decreases
in the rate of learning were correlated with higher
steady-state productivity levels and more previous
experience levels. In other words, slower learning
individuals tended to achieve higher steady-state productivity levels, while workers with more previous
experience tended to learn at a slower rate. The forgetting parameter, , was correlated with marginal
significance to prior experience, p (negatively), and
learning rate, r. Among the work history characteristics in Table 3, only the average production time
per unit was correlated (negatively) with any of the
learning/forgetting parameters, k, p, r, or . In this
case the negative correlation simply reflects the natural relationship that as unit processing times decrease,
productivity levels increase. The statistically significant correlations among the work history characteristics are similarly intuitive. For example, the correlations between days worked and both units produced
and the number of breaks suggest that as the length
of time an employee spends at an inspection station
increases, both the total quantity produced and the
number of breaks taken tend to increase.
Management Science/Vol. 47, No. 12, December 2001
SHAFER, NEMBHARD, AND UZUMERI
Worker Learning, Forgetting, and Heterogeneity
4.
The Impact of Learning and
Forgetting on Assembly
Line Performance
In addition to succinctly describing individual patterns of learning and forgetting, Equation (3) could
be used to analytically determine the performance of
a particular worker or group of workers under various scenarios. For example, the total number of radios
inspected given a precisely specified assignment and
rotation schedule could be determined using Equation (3). Unfortunately, a number of random events,
such as worker absenteesim, workers leaving the
organization, workers bumping one another based
on seniority, etc., make it extremely difficult to precisely determine total production. Simulation provides a practical modeling approach given the need
to consider the dynamic and stochastic nature of such
systems.
4.1. The Impact of Worker Heterogeneity
In investigating the impact that alternative patterns
of learning and forgetting have on the performance
of an assembly line, it becomes apparent that little
research has investigated the impact of worker heterogeneity, particularly in situations in which the workers operate independently of one another. To illustrate the nature of heterogeneity of learning rates,
consider a group of seven workers operating independently of one another such that the total output is
equal to the sum of their individual output. Further,
assume that none of the workers has any prior experience with the task and that it takes each worker an
equal amount of time (e.g., 3 hours) to complete their
first unit. We deterministically compare two scenarios, each with average learning rate of 70%. The first
scenario has each of the seven workers with learning rates equal to 70%, and the second scenario has
workers with learning rates of 66%, 68%, 69%, 70%,
71%, 72%, and 74%, respectively. Results show, counterintuitively, that the scenario with worker heterogeneity resulted in higher levels of output (425 units
versus 406 units after 40 hours), assuming no forgetting occurs. Thus, the impact of fast and slow workers is not cancelled out. Furthermore, while in many
settings increasing variation results in decreased system performance (e.g., quality and waiting lines), we
Management Science/Vol. 47, No. 12, December 2001
observe that in this situation increasing levels of variation result in improved overall system performance.
A theorem and proof more formally depicting this
phenomenon are given in the appendix. An examination illustrating the size of this difference using empirical data is given in §5.3.
4.2.
Simulating Patterns of Learning and
Forgetting on Assembly Line Performance
We now turn our attention to the specific issue of how
patterns of learning and forgetting affect the operating performance of an assembly line. In part, this
research issue was motivated by the studied organization’s interest in investigating potential benefits
associated with a new video-based technology that
delivers task-specific training material on demand to
operators at their workstations. A conceptual model
of relevant factors that influence the performance of
the assembly line is presented in Figure 1, where
the use of new technology might impact the rate of
worker learning and the propensity for workers to
forget. In addition, the length of task tenures and the
size of the worker pool are also hypothesized to have
a direct impact on performance.
To simulate the performance of the assembly line,
the set of best-fit values, k, p, r, and summarized in
Table 1, were used to create a population of 70 workers. This population was randomly ordered 10 times
to create 10 worker pools in an effort to guard against
the possibility that a nonrepresentative pool of workers was randomly chosen and used in the simulation
models. Four factors from the conceptual model in
Figure 1 were investigated and discussed below.
4.2.1. Rate of Worker Learning, r. The videobased technology may impact the rate of worker
learning, which in turn directly impacts assembly line
productivity. Analysis of the best-fit parameters for
the pool of 70 workers indicated that the workers
progressed rapidly up the learning curve. Therefore,
given the rapid rate of learning observed, the following three levels of r were included in the study: 1) the
best-fit values of r, 2) 2 × the best-fit values of r, and
3) 4 × the best-fit values of r.
4.2.2. Task Tenure, t. The amount of consecutive
time a worker is assigned to a test station, or task
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SHAFER, NEMBHARD, AND UZUMERI
Worker Learning, Forgetting, and Heterogeneity
Figure 1
Factors Influencing Assembly Line Productivity
tenure, was the second factor controlled in the computer simulation study. As shown in Figure 1, task
tenure may be directly related to patterns of learning and forgetting in two competing ways. On the
one hand, the longer workers perform a given task,
the more opportunity they have to progress along the
learning curve, resulting in higher levels of task proficiency. On the other hand, longer task tenures result
in less turnover at the workstations, thus increasing the amount of time that elapses between successive assignments to the test station for other workers.
This leads to lower performance because the length
of time between successive assignments would tend
to increase forgetting. Task tenure is also a factor
under some degree of managerial control. For example, incentives can be offered to reduce the amount of
task turnover or new work rules negotiated with the
union to reduce the amount of job bumping.
Three levels of task tenure were investigated, where
each is drawn from an exponential distribution with
1) an average task tenure of 4 days, 2) an average task
tenure of 8 days, and 3) an average task tenure of 32
days.
4.2.3. Worker Pool Size, wp. The third factor,
worker pool size, is the total number of workers available in the plant to staff the test stations. One factor
that influences the size of the worker pool is management’s attitude toward and use of cross-training programs. As shown in Figure 1, the size of the worker
pool relates to patterns of learning and forgetting in
the sense that it influences the chance that a particular
1646
worker will be selected to work at the test station each
time a vacancy occurs. More specifically, the larger
the pool of available workers, the less likely it is that
any particular worker will be selected to staff the test
station. As the likelihood of being selected decreases,
opportunities to progress up the learning curve are
fewer, and the time between successive assignments
to the test station will be greater, potentially resulting
in greater forgetting.
In the simulation study the following three levels of
the worker pool size were investigated: 1) 2 workers
available per test station, 2) 4 workers available per
test station, and 3) 8 workers available per test station.
Because the model simulated the operation of a single assembly line with eight test stations for one shift,
these three levels translate into worker pools of 16, 32,
and 64 workers, respectively. To reduce the variability across simulation runs, the first 16 workers in the
pools of 70 workers were used in all models with 2
workers per test station. Likewise, the first 32 and 64
workers were used in the models with 4 workers per
station and 8 workers per station, respectively. Any of
the workers in the worker pool could be assigned to
any station.
4.2.4. Forgetting, . The fourth and final factor
investigated was a parameter related to an individual
worker’s propensity to forget the task studied during
breaks, . As shown in Figure 1, it was hypothesized that a key benefit associated with the videobased training and information delivery system was
Management Science/Vol. 47, No. 12, December 2001
SHAFER, NEMBHARD, AND UZUMERI
Worker Learning, Forgetting, and Heterogeneity
that it would reduce the amount of worker forgetting. Three levels of were investigated: 1) the bestfit values of for the 70 workers, 2) 05 × the bestfit values of reflecting a moderate reduction in
the extent of forgetting, and 3) = 0. Setting to
zero eliminates the discounting of cumulative experience in our models since cumulative experience is
discounted by the recency measure R raised to .
Although it is unlikely that worker forgetting could
be completely eliminated, setting to zero provides
an upper limit on the potential benefits of reducing
worker forgetting.
4.3.
The Simulation Models and
Experimental Design
At the start of the simulation, the first eight workers
in the worker pool were assigned to the eight inspection stations.4 At the time of this assignment, a random task tenure was generated for each worker. After
the amount of time corresponding to the task tenure
elapsed, a worker from the worker pool was randomly selected subject to the following restrictions:
1) all workers not currently assigned to one of the
eight inspection stations had an equally likely chance
of being selected, and 2) the same worker could not
be assigned consecutively to the same station because
this would result in a longer task tenure at that station
than should occur based on the level of task tenure
specified. At the time of assignment, the task tenure
of the worker selected to fill the vacancy was randomly generated. Workstations are identical and, consequently, reassignments among stations are equivalent to staying at the current station.
Because the purpose of this study is to investigate
how heterogeneous patterns of learning and forgetting affect assembly line performance, an infinite supply of radios was available to the eight inspection stations so that these stations would never be starved
for work. This avoids confounding our results with
the amount of work in process. That is, if it were
possible for the stations to run out of work it would
be difficult to determine whether differences in performance were the result of the factors controlled in
this study or of workers missing out on opportunities
4
The simulation models were coded in Awesim.
Management Science/Vol. 47, No. 12, December 2001
to progress up the learning curve. Furthermore, since
the test stations were the bottleneck on the actual
radio assembly line, it was rare for these stations to
be starved. Performance was measured both by the
average radio inspection time and the total number
of radios inspected.
In total, 810 unique simulation models were developed (3 levels of the rate of worker learning, r × 3
levels of task-tenure, t × 3 worker pool sizes, wp × 3
levels of the factor related to worker forgetting, ×
10 different arrangements of the pool of 70 workers).
Each simulation model was run for 1 year of simulated time. A warm-up period was not used for three
reasons. First, the nature of the assembly line studied is that it is shut down once a year to incorporate changes for the new model year and then started
up again. Second, warming up the model and then
clearing the statistical arrays would bias the results
because the performance of the system would only be
assessed after the workers had gained some amount
of experience. In studies that investigate patterns of
learning and forgetting, it is precisely the transient
period that is of interest, not an equilibrium state that
may exist after workers have reached the flat part of
their learning curves. Third, the performance measures used in this study are not biased by starting
up the system empty, unlike other measures such as
average work in process. Three independent replications of 1 year were run for each of the 810 simulation
models for a total of 2,430 runs. That is, each of the
81 cells in our full-factorial experiment (3 levels of
r × 3 levels of t × 3 levels of wp × 3 levels of ) was
replicated 30 times (3 independent replications × 10
different worker pools).
5.
Results and Discussion
5.1. Limitations
Before discussing the results, we note a few limitations associated with this study. First, it should be
reiterated that employees assigned to the inspection
stations work independently. Therefore, we do not
address social factors related to individual learning
such as the impact of familiarity with other group
members. Second, because our data are limited to
1647
SHAFER, NEMBHARD, AND UZUMERI
Worker Learning, Forgetting, and Heterogeneity
Table 4
Tenure
4-day
tenure
8-day
tenure
32-day
tenure
Average Number of Radios Finished During the Year (1,000 Units)
r
2r
M
SE
95% CI
M
SE
95% CI
M
SE
95% CI
=0
16
32
64
439
427
410
3.3
1.5
1.2
432.2, 445.8
423.9, 430.1
407.5, 412.5
417
397
377
1.9
1.4
1.2
413.1, 420.9
394.1, 399.9
374.6, 379.4
393
369
347
1.5
1.4
1.1
389.9, 396.1
366.2, 371.8
344.7, 349.3
Half of
best-fit
16
32
64
431
416
399
2.7
1.3
1.1
425.4, 436.6
413.3, 418.7
396.7, 401.3
404
383
363
0.9
1.2
1.1
402.2, 405.8
380.5, 385.5
360.7, 365.3
379
356
334
1.4
1.2
1.1
376.1, 381.9
353.5, 358.5
331.7, 336.3
Best-fit
16
32
64
419
402
380
2.6
1.6
1.5
413.8, 424.2
398.8, 405.2
376.8, 383.2
393
368
345
1.2
1.3
1.5
390.6, 395.4
365.4, 370.6
341.9, 348.1
370
342
319
1.4
1.1
1.4
367.2, 372.8
339.8, 344.2
316.1, 321.9
=0
16
32
64
440
427
411
3.6
1.5
1.4
432.7, 447.3
423.8, 430.2
408.2, 413.8
418
398
378
2.1
1.4
1.4
413.7, 422.3
395.2, 400.8
375.2, 380.8
394
370
349
1.4
1.3
1.2
391.1, 396.9
367.2, 372.8
346.5, 351.5
Half of
best-fit
16
32
64
431
417
400
2.9
1.4
1.3
425.0, 437.0
414.2, 419.8
397.4, 402.6
405
385
364
1.2
1.2
1.2
402.5, 407.5
382.5, 387.5
361.5, 366.5
379
358
336
1.4
1.2
1.2
376.1, 381.9
355.6, 360.4
333.6, 338.4
Best-fit
16
32
64
418
402
383
3.0
1.8
1.7
411.9, 424.1
398.2, 405.8
379.6, 386.4
391
369
347
1.3
1.5
1.6
388.3, 393.7
365.9, 372.1
343.6, 350.4
369
343
320
1.5
1.5
1.7
366.0, 372.0
340.0, 346.0
316.6, 323.4
=0
16
32
64
438
430
420
3.9
2.2
1.7
430.1, 445.9
425.5, 434.5
416.4, 423.6
417
404
392
2.5
2.0
1.7
411.9, 422.1
400.0, 408.0
388.6, 395.4
393
377
365
1.8
2.0
1.6
389.3, 396.7
372.9, 381.1
361.7, 368.3
Half of
best-fit
16
32
64
430
422
412
3.3
2.0
1.6
423.2, 436.8
417.9, 426.1
408.7, 415.3
405
391
380
1.7
1.7
1.5
401.5, 408.5
387.5, 394.5
376.8, 383.2
380
364
353
1.5
1.7
1.5
376.9, 383.1
360.5, 367.5
349.9, 356.1
Best-fit
16
32
64
419
409
400
3.2
2.3
1.8
412.4, 425.6
404.4, 413.6
396.4, 403.6
392
377
367
1.7
2.0
1.7
388.5, 395.5
373.0, 381.0
363.6, 370.4
369
351
340
1.5
2.0
1.8
365.9, 372.1
346.8, 355.2
336.4, 343.6
Forgetting
the completion of a single task, we do not evaluate the performance impact of the type and range of
tasks assigned to workers. Third, the data collected
for this study came from a task that was partially
machine-paced and partially worker paced. Therefore,
the results may not be generalizable to situations that
are purely worker paced. Nonetheless, we call attention to the broad range of manufacturing settings to
which this study relates directly. Fourth, we note that
we do not model within-worker variability in the simulations, since this would be primarily a source of
additional noise in the context of our approach. As
a result, in practice, we would expect to see higher
variation in output than what was observed in the
simulations. Correspondingly, the variability in total
1648
4r
Size of
Worker Pool
production would be expected to be somewhat higher
in practice. Finally, while the causes for the breaks
in the work history files are unknown, their effects
are likely to show up in the best-fit parameter estimates for the workers. However, because the causes
were unknown and could result from a variety of reasons, including seniority-based bumping, vacations,
cross-training, absenteeism, maternity leaves, etc., no
attempt was made to model these causes.
5.2. Simulation Results
Table 4 summarizes the 95% confidence intervals for
the simulation experiment results in terms of the
dependent variable—cumulative inspected radios. A
small but significant dependency was created in our
Management Science/Vol. 47, No. 12, December 2001
SHAFER, NEMBHARD, AND UZUMERI
Worker Learning, Forgetting, and Heterogeneity
experiment because we used three replications for 10
worker pools at each design point. To adjust for this,
a single-factor model was used to remove the worker
pool dependency. The model accounts for the variation in the cumulative inspected radios explained by
the worker pools R2 = 347%, F = 967, p = 00001).
The residuals of this model contain the variation in
cumulative inspected radios not explained by the
worker pool number and are thus independent. Using
these residuals, the standard errors reported in Table
4 are based on the resulting 30 statistically independent samples. The 95% confidence intervals were then
calculated using t002529 .
The correlation between cumulative inspected
radios and our alternate dependent variable—average
radio inspection time—was −09939 (p value =
00001). Intuitively, this makes sense and indicates
that as the average radio inspection times decrease,
the number of radios inspected will increase. Given
this, our discussion of the results will focus primarily
on the cumulative number of radios inspected.
Table 5 provides a summary of the analysis of
variance results for average radio inspection times
Table 5
Summary of ANOVA Results for Average Radio Inspection
Times and Radios Inspection During Year
Cumulative Radios
Inspected During
Year
Average Radio
Inspection Times
Source
Model
r
t
r ×t
wp
r × wp
t × wp
r × t × wp
r ×
t ×
r ×t ×
wp × r × wp × t × wp × r × t × wp × ∗
F Value
31217
726240
20650
721
289838
10697
9259
234
159364
2295
487
010
1853
036
196
005
Pr > F
∗
00001
00001∗
00001∗
00001∗
00001∗
00001∗
00001∗
00167
00001∗
00001∗
00006∗
09992
00001∗
09429
00482
10000
F Value
Pr > F
26749
645932
15883
186
248642
3066
6673
040
136774
473
222
026
375
024
078
007
00001∗
00001∗
00001∗
01141
00001∗
00001∗
00001∗
09193
00001∗
00008∗
00646
09773
00048∗
09836
06179
10000
Denotes significant at the 0.01 level.
Management Science/Vol. 47, No. 12, December 2001
and total number of radios inspected during the year
after adjusting for the impact of the worker pool. As
shown, all four main effects were statistically significant on both performance measures. Also, four of
the six two-way interaction terms were significant at
the .01 level on both dependent variables: r × wp, t ×
wp, r × , and wp × . The other two-way interaction
terms, r × t, and t × were significant at the .01 level
only for average radio inspection times. None of the
three- or four-way interaction terms were significant
at the .01 level.
Because the interpretation of main effects may not
be meaningful when significant interaction effects are
present (Spector 1981), profile plots (Hildebrand and
Ott 1996) were created to investigate each statistically significant two-way interaction. In all the profile
plots created, there were clear main effects present
and the significant interaction terms were largely due
to relatively minor differences in the slopes of the
lines. Therefore, our discussion will focus on the main
effects, but will be qualified as appropriate to account
for the presence of significant interaction effects.
5.2.1. The Impact of Worker Forgetting, . Analysis of the confidence intervals in Table 4 indicated
that in 23 of the 27 cases, statistically more radios
were inspected going from to /2 and then further moving from /2 to = 0. In three cases statistical differences were not detected moving from /2
to = 0, and in one case statistical differences were
not detected moving from to /2 to = 0. All four
of the cases in which statistical differences were not
detected corresponded to the fastest learning environments, helping explain the significant two-way interaction term × r. Three of these cases corresponded
to environments with 16 workers, supporting the significant two-way interaction between × wp. These
results suggest that the benefits of reducing may
not be as important in faster learning environments.
Further, the benefits of reducing increase as the
size of the worker pool increases. To fully appreciate the implications of this, note that as the worker
pool size increases, the amount of time between successive assignments to the inspection station will tend
to increase. Since the workers are randomly assigned
to the inspection station when a vacancy occurs,
increasing the size of the worker pool decreases the
1649
SHAFER, NEMBHARD, AND UZUMERI
Worker Learning, Forgetting, and Heterogeneity
likelihood that any particular worker will be selected
to fill the vacancy. As the chance of being selected
decreases, the time between successive assignments
will tend to increase. Hence, the longer gaps between
successive assignments associated with larger worker
pools makes it more beneficial to reduce the negative
impacts of forgetting.
5.2.4. The Impact of the Rate of Worker Learning, r. Consistent with Figure 1, performance deteriorated as the rate of learning decreased in all cases.
Also, as was discussed previously, increasing the rate
of learning can moderate the negative impacts of
increasing the size of the worker pools and the degree
to which workers forget.
5.2.2. The Impact Worker Pool Size, wp. In all
cases the performance of the inspection operation
deteriorated as the size of the worker pool (wp)
increased from 32 to 64 workers. However, in 3 of
the 27 cases, statistically significant differences were
not observed as wp increased from 16 to 32 workers.
These results generally suggest that the performance
of the radio inspection operation improved as the task
experience was concentrated in a smaller number of
workers. The 3 cases in which statistical differences
were not detected as wp increased from 16 to 32 workers all corresponded to the fastest learning environment, helping explain the significant two-way interaction term wp × r. Furthermore, these 3 cases also
corresponded to environments with the longest task
tenures, helping explain the interaction term wp × t.
The managerial implications of these results are that
the benefits of concentrating work among a smaller
group of individuals increase as the pace of worker
learning decreases and/or the length of task tenures
decreases.
5.3.
5.2.3. The Impact of Task Tenure, t. The impact
of task tenure on assembly line performance was
mixed. In 14 of the 27 cases there were no statistical
differences in radios inspected going from 4 to 8 to 32
day tenures. However, in the other 13 cases statistical differences in total inspected radios were detected
as task tenures increased from 8 to 32 days. Of these
latter 13 cases, 9 corresponded to environments with
64 workers and 4 corresponded to wp = 32 workers,
again supporting the significant two-way interaction
term wp × t. In summary, these results suggest that
increasing the length of task tenures is most beneficial in cases with larger worker pools. Increasing the
length of task tenure did not provide any observable
benefit in the smallest worker pool environments.
1650
Follow-Up Simulations Investigating
Worker Heterogeneity
We highlighted the importance of incorporating
worker heterogeneity when modeling learning and
forgetting at the individual worker level. This previous discussion was based on insights obtained using
a deterministic example and an analytical proof. In
this section, we overview the results of a follow-up
simulation study conducted to investigate the impact
of worker heterogeneity in a more complex and realistic setting. That is, what are the likely magnitudes
of differences between modeling under homogeneous
and heterogeneous worker population assumptions?
To conduct this comparison we replace the best-fit
parameters calculated individually for each worker
with the average worker population parameters listed
in Table 2. Hence, in the homogenous simulation
models, all workers had the potential to inspect 29.7
radios per hour, had the equivalent prior experience
of inspecting 1,150.5 radios, needed to inspect 921.1
radios to reach half their potential productivity level,
and had an equal to 1.3. Given the definition of
these parameters, it appears reasonable to us that
averaging them would provide a good approximation for a typical worker. Indeed, these average values were representative of several workers’ unique
best-fit parameters. For example, one worker’s bestfit parameters k, p, r, and were 30, 1838, 623,
and 0, respectively, while another worker’s best-fit
parameters were 31.5, 1249, 927, and 0.4, respectively.
However, if averaging these four parameters does not
provide a good approximation for a typical worker,
then the question arises how should the parameters
for a typical worker be determined? The point being
that determining the parameters for a homogeneous
model composite worker is nontrivial, particularly
given the nonlinear nature of learning and forgetting.
We remark that biased results obtained as a product
Management Science/Vol. 47, No. 12, December 2001
SHAFER, NEMBHARD, AND UZUMERI
Worker Learning, Forgetting, and Heterogeneity
of not modeling worker heterogeneity may be further
compounded by the complexity of determining the
parameters for a so-called typical worker.
In the follow-up homogeneous worker case, a fullfactorial experiment was conducted with the same
four factors included in the heterogeneous case.
Because one of our purposes is to investigate the
impact of not modeling worker heterogeneity, only
the two extreme levels were used for each of these
four factors. Therefore, our follow-up experiment
included 32 design points: 2 levels of r (r and 4r) × 2
levels of t (4 and 32 days) × 2 levels of wp (16 and
64 workers) × 2 levels of (zero and best-fit values)
× 2 levels of the learning curve parameters specified
for each worker (individual best-fit parameters and
average of best-fit parameters). The 16 design points
where the best-fit parameters are used results in the
same models used in the main study, and therefore
these results were carried over to the follow-up study.
The models where the average population parameters
were used in place of each worker’s unique bestfit parameters had to be developed for this followup study. It is worth noting that in the models with
the average values of the parameters, all workers are
identical and therefore it is not necessary to reorder
the workers to create different worker pools. In a similar fashion to the heterogeneous main experiment,
the simulation models in the follow-up experiment
were independently replicated 30 times, each replication representing one year of simulated time, and no
warm-up period.
Table 6 summarizes the results of our comparison
between the models that use each worker’s unique
best-fit parameter values and the models that use the
average of the best-fit parameter values for all workers. Included in the table is the percentage change in
mean and standard deviation of the number of radios
inspected. Consistent with our earlier discussion, in
all cases a smaller number of radios was inspected in
the homogeneous case than the heterogeneous case.
The range of the difference was from 2.0% to as much
as 30.6%. Furthermore, not only did using the best-fit
parameters result in higher levels of finished product, it also resulted in much higher levels of variation
across the simulation runs. These results are consistent with the results discussed earlier and suggest that
Management Science/Vol. 47, No. 12, December 2001
Table 6
Percentage Change in Average Number of Radios Inspected
(BF − Avg)/Avg
Forgetting
=0
16
64
Best fit
4-day Tenure
Worker
Pool Size
16
64
M
SD
M
SD
M
SD
M
SD
32-day Tenure
r
4r
r
4r
25
536256
95
5971
58
41353
143
5185
163
71130
276
3500
306
18017
207
43409
20
45370
59
1738
64
141
116
1100
148
5460
227
810
274
2318
260
3006
Note. BF= models were each worker’s unique best fit-parameters were
used. Avg= models were same average parameter values were used for all
workers.
using a single learning curve could significantly bias
productivity levels downward while at the same time
greatly underestimating the amount of variation associated with the results. An important implication of
this is that calculated confidence intervals would be
too narrow given the level of significance specified.
6.
Conclusions
The purpose of this study was to investigate how heterogeneous patterns of learning and forgetting affect
the performance of an assembly line. In the course of
addressing this specific issue a more general research
issue emerged. More specifically, what is the impact of
modeling situations where workers operate independently of one another with a single composite learning curve?
Regarding the impact of using a single composite
learning curve for all workers versus using individual learning curves for the workers, the results of this
study clearly demonstrate that not modeling inherent variations across workers can lead to significantly
underestimating overall productivity in environments
where the workers operate independently. This somewhat nonintuitive result was observed using deterministic simulation based on the one-parameter loglinear model and varying the learning rate across
a population of workers while holding the average
learning rate constant. We also verified this result analytically by demonstrating that a group of workers
1651
SHAFER, NEMBHARD, AND UZUMERI
Worker Learning, Forgetting, and Heterogeneity
who are heterogeneous with respect to their learning parameters will outperform a group made up
of “average” workers characterized by mean parameter estimates. In our simulations of more practical settings, we observed that across the environments studied, system output was underestimated
by an average of 15.3% when a homogeneous workforce was assumed with a range of 2% to over 30%
underestimation.
In terms of how patterns of learning and forgetting affect the performance of an assembly line, the
results of this study provide support for the conceptual model shown in Figure 1. More specifically,
the results suggest that increasing the rate of worker
learning, perhaps via the adoption of new technologies, can mitigate the negative impacts of both larger
worker pools and worker forgetting. Furthermore,
and perhaps counterintuitively, our results suggest
that increasing the length of task tenures does not
automatically translate into higher levels of system
performance. In the cases investigated here, longer
task tenures did not provide any observable benefit
in the environments with the smallest worker pools.
There are a number of ways this research can
be extended. For example, in the present study, the
workers were never starved for work. Future research
may investigate the trade-offs between shops with
higher levels of congestion and the confusion this creates, with increased opportunities to progress up the
learning curve due to never being starved for work.
A number of important issues were discovered
in the course of analyzing the results of this study.
For example, having determined the importance of
modeling worker heterogeneity, additional research
is needed investigating how the parameters of the
distribution of the learning curves affects shop performance. In the present study only two of the
parameters, those that describe rates of learning and
forgetting, were investigated. Future studies should
investigate how the other parameters affect shop performance. For example, while we dealt with the issue
of turnover at the individual task level in this study,
the parameter p relates in a similar way to employee
turnover at the organizational level, which also has
clear managerial implications.
1652
Research is also needed to further investigate how
worker heterogeneity impacts system performance. It
would be worthwhile to investigate under what circumstances it is reasonable to use a single learning
curve versus situations where it is more appropriate
to use individual learning curves. Additional research
is needed to determine procedures for calculating the
parameters for the learning curve in situations where
it is appropriate to use a single curve. For example, what impact does the distribution (including both
variation and shape) of the parameter estimates have
in estimating composite parameter values? In a sense,
it was the current research approach that allowed
us to identify these important issues as we hope to
address many of them in the future.
Acknowledgments
The authors thank the associate editor for her helpful guidance and
the three anonymous reviewers for their thoughtful and insightful
suggestions. This research was supported by the Babcock Graduate
School of Management, Wake Forest University Research Fellowship Program, and The University of Wisconsin-Madison Graduate
School Research Grant.
Appendix
The following theorem and corresponding proof demonstrate
that greater variation among individual worker learning function
parameters results in higher overall productivity. That is, basing
simulations (or perhaps other analyses) on an average worker will
tend to underestimate productivity. This theorem applies to a broad
class of learning functions, where one could reasonably expect similar behavior for models of both learning and forgetting, since typically no work takes place during the periods where forgetting takes
place.
Definitions. Let f t represent a learning function that is in
the general class of functions, which are increasing and concave
down for production rate (units/time). Without loss of generality,
we remark that functions for production time (time/unit) are reciprocal in nature and correspondingly decreasing and concave up.
Further let
t = elapsed (cumulative) time in hours since start
of production;
UT x = number of units produced during time T given
a set of learning parameters x and
x = x1 x2 xn Management Science/Vol. 47, No. 12, December 2001
SHAFER, NEMBHARD, AND UZUMERI
Worker Learning, Forgetting, and Heterogeneity
Theorem. In a concave learning function of time versus production
rate, f t (units/time), average production with heterogeneity is greater
than the production of an average learner. That is, if Var
x > 0, then
1
U xi > U x̄
n i
(A-1)
Proof. To determine the production during time T , we integrate
the instantaneous production rate function f t as follows:
UT x =
0
T
f t dt = F T − F 0
(A-2)
Since f t is increasing and concave down, F ’
T > 0, F T > 0,
with F 0 a constant, from which we conclude that UT is concave
up. It is then a straightforward application of Jensen’s inequality
(Weisstein 1999) that for UT concave up, and Var x > 0, that Equation (A-l) holds true, thus showing the desired result. References
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1653