The Effects of Worker Learning, Forgetting, and ... - Semantic Scholar

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

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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 25
2 3
20 5
60 3
9 11
43

1496 16
3 2
19 4
94 2
2 8
91

7719 69
1 14
82 12
80 10
0 21
51

2266 23
5 2
37 4
36 3
0 9
60

Min
364 3
0 1
03 1
72 0
0 4
83

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)



29
7 7
4 45
0 19
4

1150
5 2496
1 13165
7 0
0

921
1 2289
3 14980
9 1
3

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 0
505 r 0
658∗∗ 0
827∗∗  0
086 −0
249∗ 0
244∗ Units produced 0
128 0
117 0
086 0
204 No. breaks 0
030 0
054 0
013 0
177 0
639∗∗ Total days worked 0
030 0
025 0
014 0
111 0
849∗∗ 0
557∗∗ (excluding breaks) Task tenure 0
103 0
017 0
050 −0
022 0
281∗ −0
258∗ Break length (days) −0
121 −0
160 −0
137 −0
073 −0
207 −0
225 Average production −0
269∗ −0
117 0
031 −0
159 −0
464∗∗ −0
337 time per unit (minutes) ∗

0
560 −0
158 −0
338∗∗

0
015 −0
009

−0
127

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 1645

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 312
17 7262
40 206
50 7
21 2898
38 106
97 92
59 2
34 1593
64 22
95 4
87 0
10 18
53 0
36 1
96 0
05

Pr > F ∗

0
0001 0
0001∗ 0
0001∗ 0
0001∗ 0
0001∗ 0
0001∗ 0
0001∗ 0
0167 0
0001∗ 0
0001∗ 0
0006∗ 0
9992 0
0001∗ 0
9429 0
0482 1
0000

F Value

Pr > F

267
49 6459
32 158
83 1
86 2486
42 30
66 66
73 0
40 1367
74 4
73 2
22 0
26 3
75 0
24 0
78 0
07

0
0001∗ 0
0001∗ 0
0001∗ 0
1141 0
0001∗ 0
0001∗ 0
0001∗ 0
9193 0
0001∗ 0
0008∗ 0
0646 0
9773 0
0048∗ 0
9836 0
6179 1
0000

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

2
5 53625
6 9
5 597
1 5
8 4135
3 14
3 518
5

16
3 7113
0 27
6 350
0 30
6 1801
7 20
7 4340
9

2
0 4537
0 5
9 173
8 6
4 14
1 11
6 110
0

14
8 546
0 22
7 81
0 27
4 231
8 26
0 300
6

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 x  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.


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Accepted by Linda Argote; received December 1, 1998. This paper was with the authors 12 months for 3 revisions.

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