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Modeling simultaneous worker learning and forgetting in dual resource constrained systems
European Journal of Operational Research 115 (1999) 158±172
Theory and Methodology
Modeling simultaneous worker learning and forgetting in dual resource constrained systems Hemant V. Kher a, Manoj K. Malhotra b
b,*
, Patrick R. Philipoom b, Timothy D. Fry
b
a Division of Business and Economics, University of South Carolina Sumter, 200 MillerRoad, Sumter, SC 29150-2498, USA Department of Management Science, The Darla Moore School of Business, University of South Carolina, Columbia, SC 29208, USA
Received 1 July 1997; accepted 1 April 1998
Abstract This paper addresses issues related to modeling worker learning and forgetting eects in Dual Resource Constrained (DRC) systems. The learn-forget-learn (LFL) model of Carlson and Rowe (Carlson, J.C., Rowe, A.J., 1976. Industrial Engineering 8, 40±47) is used to critically evaluate several worker training related issues that are associated with the learning and forgetting phenomenon. A numerical analysis is performed on the LFL model within a DRC system to gain insights into the nature of relationships between the extent of worker ¯exibility, forgetting rates, attrition rates, and ¯exibility acquisition policies. Results suggest that in the presence of higher attrition and forgetting rates, a worker may not be able to achieve full eciency in as little as two dierent departments. Thus acquiring even incremental worker ¯exibility under such conditions may be infeasible. We also show that managers can use dierent ¯exibility acquisition policies designed in this study to reduce relearning losses and consequently improve the system performance. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Numerical analysis; Modeling; Scheduling; Dual-resource constrained systems
1. Introduction In response to an increased emphasis on lower inventories, on-time deliveries, and higher and consistent product quality, operations managers are striving to improve their manufacturing processes. One such method for process improvement is the development of ¯exible resources, in partic-
*
Corresponding author. Tel.: 001 803 777 2712; fax: 001 803 777 6876; e-mail: [email protected].
ular workforce ¯exibility (Malhotra and Ritzman, 1990). Flexible workers can provide a buer against uncertainties that are inherently present in any manufacturing system such as machine breakdowns, product mix changes, external demand changes, and material shortages. A ¯exible workforce also helps in reducing work-in-process inventory levels, thereby allowing for improvements in manufacturing lead times and customer service performance (Treleven, 1989). Japanese ®rms such as Mitsubishi Belting and Matsushita Electric are known to encourage worker ¯exibility
0377-2217/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 8 ) 0 0 1 9 0 - 8
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by tying wages directly to the dierent set of skills possessed by the workers (Schonberger, 1982). Similarly, studies by Rohan (1987) and Kertesz (1987) indicate that ®rms like Frito Lay and General Motors have adopted worker ¯exibility to gain from its bene®ts. Though bene®ts associated with a ¯exible crosstrained workforce are well understood, the same cannot be said for the cost of acquiring and utilizing worker ¯exibility. The costs occur in part during the cross-training period due to worker learning eects, which result in productivity losses since the workers are typically less ecient during training. The losses during the learning phase can be further exacerbated if interruptions occur due to varying workloads between departments causing workers to leave an under loaded department before training is complete. The ongoing use of worker ¯exibility after complete training has occurred may also be expensive if prolonged absence from a given department leads to worker forgetting. Similarly, attrition can also add to learning losses if the new worker needs training in order to be utilized ¯exibly. Thus acquiring worker ¯exibility can result in a one time training loss, while forgetting and relearning represent the ongoing cost of using it. Bobrowski and Park (1993) give speci®c examples of ®rms incurring training related losses (Chrysler corporation used 900,000 h before launching the new Jeep, and GM provided each worker over 80 h of training before starting new product lines (p. 258)). Similarly, Wisner and Pearson (1993) report that ®ve of the six DRC shop managers they surveyed for their study attested to the presence of worker forgetting when transfers took place, and that three of these managers used frequent periodic training to control their relearning problem. The preceding discussion suggests that both learning and forgetting (which leads to relearning) phenomenon must be modeled and studied simultaneously in order to truly understand how worker ¯exibility should be acquired and used to provide ongoing bene®ts to the ®rm. Such an understanding becomes especially critical for a large number of ®rms that are known to rely on labor
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productivity as a primary measure of shop performance (Fry et al., 1995b). In such ®rms, the productivity losses attributable to worker learning and relearning may be a signi®cant source of concern for measuring performance and setting standards. In this paper, we perform numerical analysis on a well known worker learning and forgetting model proposed by Carslon and Rowe (1976). This is done in the context of understanding several training related issues in the management of dual resource constrained (DRC) systems. Given the detrimental eect that productivity losses can have on inventory and customer service performance of a DRC ®rm, managers must carefully plan the process of acquiring and operationalizing ¯exibility in the presence of learning and relearning eects. Important issues that these managers must address include deciding the number of dierent tasks for which the workers should be trained (termed as the degree of ¯exibility), how to train the workers, and how to assign workers to dierent tasks during the crosstraining period so as to maximize learning and minimize the relearning losses. Addressing these training and worker assignment issues in the simultaneous presence of learning and relearning will facilitate a better understanding of how many dierent tasks a worker can eectively learn in dierent types of learning and forgetting environments. A review of the literature on the modeling of learning, forgetting, and relearning is presented in Section 2 to justify the choice of the learn-forgetlearn (LFL) model that is used to capture these eects. Managerial policies related to the ¯exibility acquisition process are discussed in Section 3. A brief description of the one worker DRC system which serves as the vehicle for the numerical analysis of the LFL model can be found in Section 4. The performance measures associated with the numerical analysis are discussed in Section 5. We discuss the details of the LFL model as implemented in our DRC system in Section 6. Results of the analysis are presented in Section 7, and the conclusions of the study are ®nally presented in Section 8.
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2. Modeling learning and relearning The consideration of learning eects when setting labor standards has been well documented. Even though several learning models have been oered over time Yelle (1979), the log-linear model of Crawford (1944) has been used most frequently in a variety of studies to capture the learning effects. In this model, unit processing times improve at a uniform rate as the cumulative number of units produced by the learning workers is doubled. The rate at which workers learn, the time required to fully learn new tasks, as well as the associated labor productivity losses in a DRC environment are a function of several dierent factors. These factors include the nature of task being performed (e.g. machine or labor intensive tasks (Hirsch, 1956), planning of the training programs, training facilities, monetary incentives oered to the learning workers during the training period, etc. (Balo, 1970, 1971; Yelle, 1979)). As these factors are not constant across dierent industries, the learning rates, the length of the training period, and the labor productivity losses can be expected to vary. In situations where fully trained workers reach an ``ideal'' standard processing time and no further improvement is possible, a plateau log-linear learning model is more appropriate. Such a condition would readily exist when machine capability ultimately determines unit processing time, and has been observed in industry (Balo, 1966; Yelle, 1979). The plateau log-linear model has been previously used in DRC studies (Malhotra et al., 1993), and is shown as ÿ ÿL Yn1 Max U1 Xn1 ; 1
1
where Yn1 is the time required to process the next job, U1 the initial processing time factor, L a constant determined by log10 (learning rate)/ log10 (0.5) and Xn1 is the next job to be processed. In contrast to learning, the literature on the modeling of relearning is not extensive, although several studies have acknowledged its existence in practical settings (Bailey, 1989; Globerson, 1980; Globerson et al., 1989). Bailey (1989) investigated the relearning phenomenon in a laboratory setting,
and using a single break in the learning process showed that forgetting of a task depends on the nature of tasks being performed (continuous control versus procedural tasks), forgetting rate, amount learned prior to the interruption, and the length of the interruption interval. Similarly, studies by Globerson (1980) and Globerson et al., (1989) also used a single break in the learning process, and indicated that the power curves similar to that shown in Eq. (1) provide the ``best ®t'' in modeling relearning. These studies also showed that learning after interruption followed the traditional learning pattern, and that the initial processing time for the ®rst job to be processed after the interruption was a function of the length of the interruption interval as well as prior experience. Forgetting eects have been estimated in the existing literature by using both subjective guesses (Anderlohr, 1969), as well as mathematical models (Homan, 1968). Among the available approaches, the learn-forget-learn (LFL) model proposed by Carslon and Rowe (1976) has been used to capture forgetting eects in several dierent lot sizing studies (e.g. Adler and Nanda, 1974a, b; Sule, 1978). This log-linear model assumes that when an interruption occurs during learning, the worker will move back up on the learning curve. The exact point to which experience deteriorates depends on the amount of experience gained prior to the interruption, the forgetting rate, and the length of the interruption interval. The extent of forgetting is thus captured using a power model, and is consistent with the ®ndings of Globerson et al. (1989). The LFL model determines the processing time for the n 1th job in accordance with Eq. (1). However, the resumption point after every interruption is calculated as h i ÿF =L 1; 2 Xn1 Xn 1F =L Xn e where Xn1 is the ®rst job to be processed after the interruption interval, Xn the number of jobs processed thus far, F is a constant determined by log10 (forgetting rate)/ log10 (0.5) and e is the number of jobs that could have been processed during the interruption interval. In Eq. (2), the term e is calculated by determining the eciency of the worker at the point of
H.V. Kher et al. / European Journal of Operational Research 115 (1999) 158±172
interruption. Speci®cally, if the interruption occurs when the worker is processing jobs at the ideal standard rate (i.e. with complete eciency), then the number of jobs that could have been processed by the worker during the interruption interval is computed by simply dividing the interruption interval by the standard processing time. However, if the worker is interrupted during the learning period, then the actual number of jobs that could have been processed during the interruption interval is determined as if learning had continued during the interruption interval. Thus the LFL approach takes into account the exact level of experience gained by the worker just before interruption occurs. In contrast, Wisner and Pearson (1993) modeled the relearning phenomenon very dierently from the prior research and the LFL model described above. Relearning rates of 0%, 30%, and 60% were studied, where a 30% relearning implied that the ®rst job processed by a worker after a transfer has a processing time of 130% of the standard processing time. Further, as the consecutive jobs processed at a machine doubled, the processing time decreased at a rate of 70%, thereby implying a 70% rate of learning after the interruption. Thus their model diers from the LFL model in two important ways. First, the amount of prior experience possessed by that worker is ignored. Every interruption is treated the same, irrespective of the workers prior experience. Secondly, the length of the interruption interval is also ignored in determining the extent of relearning losses. Our own experiences with actual industrial situations indicate that due to all the factors considered, the LFL model more accurately depicts the forgetting and relearning process in practical settings than the Wisner and Pearson (1993) model. In particular, we cite the situation at a plant that manufactures automatic switching systems. The learning takes place with respect to elaborate setups at machines. Sometimes, these setups can well exceed the actual processing time of the job that uses that setup. Workers can take as long as six months to become pro®cient at a machine, and then forget that skill when transferred to another department in the shop. Return to the
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same department does not start at full pro®ciency, but is dependent upon how long the operator had spent at that machine in the past before the transfer occurred. The forgetting and relearning still takes place even after multiple breaks of this nature, provided the task itself is of sucient complexity and takes a long time to master in the ®rst instance. How best to train and ¯exibly deploy the workers in such ®rms is not known, since the LFL model has not been previously examined in dual resource constrained settings. The best policies may lie in minimizing the loss of eciency that will inevitably result when a worker learns and forgets too much due to a transfer to another department. The degree of learning loss will depend upon the length of absence, and transferring a worker may not even be worthwhile if relearning will severely diminish worker productivity. Yet, not transferring a worker at all forgoes the well documented bene®ts associated with worker ¯exibility (e.g. Fryer, 1974b; Treleven, 1989; Malhotra et al., 1993). These trade-os, which primarily spring from the policies used to train and deploy workers within the shop, must be better understood in order to further our knowledge of worker ¯exibility in DRC systems. 3. Managerial policies and operating environment The best set of policies that can be used to train workers to acquire the desired level of ¯exibility in DRC systems, as well as the worker assignment policies that can be used to reduce the relearning eects, are not immediately apparent. We propose that managers can acquire the desired level of ¯exibility through dierent ¯exibility acquisition policies (or training schemes). We will de®ne such policies here, and test them by conducting a numerical analysis of the LFL model across a wide range of conditions that characterize the operating environment of a DRC system. The relevant operating factors would include the attrition rate of workers and level of ¯exibility acquired. It could be argued that giving more training to workers under conditions of high learning and forgetting losses may not be worthwhile if they are going to
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leave the system due to attrition before the bene®ts of their training have been fully realized. 3.1. Worker attrition Worker attrition directly impacts the performance of DRC shops since newly hired workers require greater training than workers who have been in the shop for a period of time. In other studies (Fry et al., 1995a; Malhotra et al., 1993), worker attrition has been modeled simply as a proportion of workers lost per year. Modeling attrition in this fashion does not capture its true impact, since even a small rate of attrition may be signi®cant if the time required to train a worker is relatively long. In order to better capture the relationship between attrition rate and the time required for training, we use the ratio between attrition interval and the time required for training. Here the attrition interval is simply the mean time between consecutive attritions. Recognizing this relationship represents a major improvement in the study of worker ¯exibility in DRC shops. We model seven dierent worker attrition interval to training period ratios (atr/tr) of 2, 4, 8, 12, 16, 20, and 40, respectively. A ratio of 2 implies that the worker attrition interval is twice as large as the training period and represents the harshest setting, while larger ratios represent relatively milder attrition environments. These attrition interval ratios can also be converted into an average annual attrition rate. Consider for example a DRC system which has 12 workers. If it takes 4 weeks to train a worker, then an atr/tr ratio of 2 implies that the shop will lose 6.25 workers per year (50 weeks), thereby implying a 52.08% annual attrition rate. Similarly with ratios of 4, 8, 12, 16, 20 and 40, the corresponding annual attrition rates in the above example would be 26.04%, 13.02%, 8.68%, 6.51% 5.20%, and 2.60%, respectively. 3.2. Flexibility acquisition policies As used in this study, the ¯exibility acquisition policy represents the manner in which workers are trained. One such policy is to move workers after
each job is completed, thereby training workers simultaneously in several dierent departments. Another widely divergent policy may be to train workers in only one department for an extended period of time before allowing them to begin training in a new department. These two policies represent the two extremes between the desire for greater ¯exibility right away and the desire for greater worker eciency or productivity over the long run. We introduce three ¯exibility acquisition policies (FAP) to represent this tradeo. The ®rst level (FAP-0) involves training a worker to perform tasks in the desired number of departments with no restrictions placed on the transferring worker. The worker is transferred to the next eligible department to perform a task after processing a given batch size, and before complete learning has taken place in the current department. This approach allows the DRC system to better utilize worker ¯exibility during the cross-training period by frequent transfers, but would also likely increase the relearning eects and decrease worker productivity. At the second level (FAP-1), the worker is not allowed to transfer until the ideal standard is reached, while FAP-2 requires the worker to train in one department for twice the number of jobs required to reach the ideal standard. FAP-2 thus represents a conservative policy that fosters a greater level of worker eciency. It could be used when the management does not want to incur large relearning losses during the initial learning periods when ¯exibility is being acquired. 3.3. Worker transfer policies The worker training policies mentioned above control only the timing of the very ®rst transfer for a worker from a given department. Once the desired level of training has been provided in that department, transfer policies are used to determine ``when'' and ``where'' to transfer the worker. The ``where'' rule has been shown to be a less important determinant of performance than the ``when'' rule. Hence it is ®xed in this study by sequentially alternating a worker in the eligible departments of
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the DRC system modeled. The ``when'' rule has been shown to have a signi®cant impact on the DRC shop performance by a large number of studies (see e.g. Treleven, 1989). It essentially de®nes the frequency of worker transfers. More frequent transfers lead to a more ¯exible use of the workforce, but also increase the number of interruptions in the learning process, thereby reducing worker eciency due to an increase in the relearning losses. In our DRC system, the ``when'' rule is implemented by using the concept of batch size. The batch size is also the parameter used in transferring the worker between departments during the FAP-0 training scheme. As an experimental factor, batch size is set at three dierent levels starting with a low of 10 jobs to a high of 250 jobs. At the low end, the worker would spend at least 10 time units in a department (after the training scheme has been implemented), and perhaps longer if the ideal standard has not been reached. Given the transfer time penalties associated with forgetting, it was not considered reasonable to process less than 10 (or more than 250) jobs in a given department. In general, smaller batch sizes will reduce the worker's stay in any department and also lead to more interruptions in the learning process. On the other hand, while larger batch sizes appear to restrict bene®ts of ¯exibility, they will also reduce relearning losses. The dierences between dierent ¯exibility acquisition policies will also diminish with the use of larger batch sizes, since the batch policy and ¯exibility acquisition policy will interact with each other during the training phase of the workers. Once complete training has been achieved, batch sizes will determine the actual amount of ¯exibility utilized.
¯exibility of two and three respectively, and were chosen from a prior study which suggests that training workers to perform tasks in more than three dierent departments in the presence of high learning losses can considerably worsen system performance (Malhotra et al., 1993). Learning and relearning eects during the cross-training period are captured for the worker by using the LFL model (Eq. (2)). Based on prior literature (Malhotra et al., 1993), an 85% learning rate is selected with the initial unit processing time in a department being 400% of the standard time. These parameter values have been shown in this earlier study to represent challenging learning environments with complex tasks. We are less interested here in modeling simpler learning situations where the forgetting and relearning phenomenon is not signi®cant enough to warrant management's attention. Consequently, we also choose two different forgetting rates of 85% and 95% to represent a reasonably challenging and wide range of forgetting conditions. Finally, the worker's learning and forgetting are not department speci®c. The numerical analysis proceeds by training the worker to process jobs in two or three dierent departments, depending upon the degree of ¯exibility being investigated. Each department is assumed to have a sucient number of jobs waiting to be processed such that it is never starved for work. The worker is sequentially assigned to different departments after processing the given batch size of jobs in each department. The total number of cycles executed by the worker depend upon worker attrition, which is modeled using the exponential distribution. The attrition interval which was described in the preceding section serves as the mean value for this distribution. The probability of attrition is computed at the completion of each cycle using>
4. The DRC system
CPn
Worker training and deployment related issues are investigated by performing a numerical analysis of the LFL model in the context of a single worker in a DRC system who is trained to process jobs in two or three functionally dierent departments (or machines). These represent levels of
n X P atri ; i1
P atri eÿFTi =INT ÿ eÿSTi =INT
8 i 1; . . . ; n; 3
where CPn is the cumulative probability of worker attrition over the ®rst n cycles P(atr)i the probability of attrition during the ith cycle, STi the start
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time of the ith cycle, FTi the completion time of the ith cycle and INT is the length of the attrition interval (mean attrition inter-arrival time). The cycle is chosen as a unit of analysis since it permits the division of the attrition interval into smaller discrete portions. The probability of worker leaving within each cycle is computed based upon the selected attrition rate, and added to obtain the cumulative probability of attrition. Analysis of performance for a given set of conditions is terminated when this cumulative probability of worker attrition equals unity.
processing time of one unit by the actual processing time of the worker at the end of the analysis. This de®nition of labor eciency is also consistent with that used in prior literature (e.g. Adam and Ebert, 1992, p. 12). If the forgetting rates are high, it is quite possible that even though the system has reached steady state, a worker may not become fully ecient before leaving the system. In that case, the worker's steady state processing time at the end of the analysis will be greater than the standard processing time of one time unit, and the ®nal level of eciency will be less than 100%.
5. Performance measures
6. Implementation of the model
The objective behind the analysis is to understand the learning and forgetting process in the presence of attrition, and how ¯exibility acquisition and worker assignment policies aect performance when managers attempt to attain dierent levels of worker ¯exibility. As such, we use two criteria to assess these impacts. These two criteria are the average job processing time for the tenure of employment for the worker (APT), and the ®nal eciency of the worker. The APT measure is computed as
The steps involved in performing a numerical analysis of the LFL model in a DRC setting for a single worker are explained here. The DRC system where the worker is trained to process jobs in two or three functionally dierent departments is illustrated in Fig. 1. In conducting the numerical analysis, it is necessary to determine the batch processing time (BPT) in a given department for a given batch size. In addition, it is also necessary to determine the quantity e, which represents the number of jobs that could have been processed in the interruption interval if the learning were to continue. The quantity e is then used in Eq. (2) to determine the resumption point from where learning is to resume for the worker in a given department. Note that since the forgetting rate (F) is a continuous variable, the cumulative number of jobs completed is not always an integer quantity. Consequently, the starting and ending job numbers (which represent the batch size that is processed) over which BPT is to be determined is also not guaranteed to always be an integer quantity. Similarly, the quantity e also may not always be an integer number in general. Fig. 2 is useful in understanding the equations used to compute the terms BPT and e. Case 1 represents the situation where the processing of a batch begins and ends on the slope of the learning curve. Case 2 represents the situation where the processing of a batch begins on the slope, but ends on the plateau region of the curve. Finally, Case 3
APT
n X i1
P atri FTi ÿ STi =Ni n;
4
where APT is the attrition adjusted average processing time over all n cycles, Ni the number of jobs processed during the cycle i and n is the total number of cycles. The APT measure is computed at the end of each cycle by multiplying the average processing time for each job during that cycle (length of the cycle/number of jobs processed during the cycle) by the probability of attrition during the cycle. When the cumulative probability of worker attrition approaches unity, the sum of attrition adjusted processing times is divided by the total number of cycles to determine the attrition adjusted average processing time (APT). The second criterion used to measure performance is the ®nal level of eciency achieved by the worker. It is computed by dividing the standard
H.V. Kher et al. / European Journal of Operational Research 115 (1999) 158±172
165
Fig. 1. Schematic of the DRC shop for numerical analysis.
represents the situation where the processing of a batch begins and ends on the plateau. Expressions for computing BPT for each of the three cases are derived as shown below. It must be noted here that BPT is the dierence (FTi ) STi ) that was used in Eq. (4) to compute the APT. Case 1: Slope to slope XZ BS
U1 X
BPT h
ÿL
Case 2: Slope to plateau. This expression is derived by dividing the integral into the slope and plateau regions Zb
XZ BS
dx
X
b
BS ÿ b ÿ X ;
X
5
where X is the very ®rst job number in the current batch, BS the batch size and other terms (U1 and L) are de®ned as before.
1 dx
h i ÿL1 ÿL1 fU1 = 1 ÿ Lg b 1 ÿ X 1
dx fU1 = 1 ÿ Lg
i X BS 1ÿL1 ÿ X 1ÿL1 ;
U1 X
BPT
ÿL
6
where b is the intersection of the slope and the plateau regions of the learning curve. Case 3: Plateau to plateau XZ BS
1 dx BS:
BPT X
7
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Fig. 2. Illustration of Cases 1, 2, and 3 for batch processing time and interruption interval calculations.
Expressions for computing e, the number of jobs that could have been processed during the interruption interval if learning were to continue, are shown below. Case 1: Slope to slope h i1ÿL ÿL1 ÿ X ÿ 1; e fIT 1 ÿ L=U1 g X 1 8 where IT is the length of the interruption interval expressed in time units. Case 2: Slope to plateau h i ÿL1 ÿL1 ÿ X 1 e fU1 = 1 ÿ Lg b 1 b ÿ X:
9
Case 3: Plateau to plateau e IT:
10
In order to better understand the implementaion of our DRC system, consider an example situation where the worker is being trained to process jobs in two dierent departments. In this case, a cycle is completed when the worker sequentially
processes the predetermined batch size in each of the two departments (i.e. department #1, and department #2). For the very ®rst visit in a given department, the batch size to be processed is the maximum of the quantity speci®ed by the ¯exibility acquisition policy or that used for transferring the worker. Thus in the setting where FAP-1 ¯exibility acquisition policy is used and the worker is transferred using a batch size of 250, the worker must process 370 units at the very ®rst visit in each of the two departments. If the FAP-0 policy were to be used rather than FAP-1 in the above example, then the worker would be required to process 250 jobs in each of the two departments. For the subsequent cycles, the worker is required to process the number of jobs speci®ed by the batch size which is used to make the transfer decision (i.e. 250 jobs in the above example). The speci®c point from where the worker resumes learning in a given department is calculated using Eq. (2). The value of e in Eq. (2) is given either by Eq. (8), or by Eq. (9), or by Eq. (10) depending upon the exact case under consideration. The existence of any case scenario 1±3 depends upon the worker's prior
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experience in the department, the length of the interruption interval, and the forgetting rate. The two performance measures are collected at the end of ®nal cycle for the worker, whereby the total number of cycles (n) executed by the worker are dependent upon the existing worker attrition rate. The ®nal eciency of the worker is computed when the worker departs the DRC system. The second performance measure of APT for any given policy (or experimental cell) under consideration is obtained by using Eq. (4), with the requisite expression for BPT being given by Eqs. (5)±(7). This model was implemented through a Fortran program written on a UNIX based IBM RISC system. The model code was validated by matching its results with calculator based computations of the relevant equations carried out for small size example problems. In all cases examined, the numerical results obtained through the two methods were identical. This allowed us to execute the experimental design described earlier. A single observation of the two performance measures for each experimental cell was obtained. No statistical analysis of the data was needed since the speci®cation of the LFL model used does not include any random variations. These results are discussed in the next section.
7. Results Table 1 shows the results for dierent performance measures for the FAP-0 and FAP-1 policies, while the graphical illustration of these results can be seen in Fig. 3. Based on Eq. (1), a worker reached the ideal standard for the 85% learning environment after processing 370 jobs consecutively in a given department for the FAP1 policy. In contrast, the worker in the FAP-2 policy was not allowed to transfer until 740 consecutive jobs had ®rst been processed in the department in which learning was taking place. The trend of the results for the FAP-2 policy were similar to the ones for FAP-1 policy, even though the average eciency of the worker during the learning period increased as we progressed towards the FAP-2 policy. Hence the results for
167
FAP-2 policy are not shown here in Table 1. However, we do subsequently identify the conditions under which managers should use FAP-2 acquisition policy to exploit the bene®ts of a trained worker. As can be seen in Fig. 1 for the FAP-0 policy, the APT for the incremental ¯exibility case (F 2) is always smaller than the APT for F 3. However, the dierences between the APT for these two ¯exibility levels decrease with a reduced forgetting rate (FR 95%). Table 1 shows that except for the highest attrition rates (atr/tr ratios of 2 and 4) and forgetting rate of 85%, a worker can be fully trained in two dierent departments (F 2) under all the other conditions tested as evidenced by ®nal eciency. In contrast, complete pro®ciency for ¯exibility of 3 can only be reached for high forgetting rates (85%) with very large batch sizes. Table 1 also shows that a worker's level of experience increases with increase in batch size. This in turn implies that as the batch size is increased, the worker is able to become more ecient as measured by APT and ®nal eciency. As expected, a reduction in forgetting rate (going from FR 85% to FR 95%) has a bene®cial impact on the worker performance. For a ®xed batch size and atr/tr ratio, the APT of the worker under the 95% forgetting environment is always better than the performance under the 85% forgetting rate (see Table 1). Performance also improves with increase in the atr/tr ratio, which causes the worker to stay in the shop for a longer period of time. This in turn allows him to become more ecient and improve the APT. Providing more training on one machine before transferring, as controlled by FAP-1 and FAP-2 policies improves performance. Results show that when acquiring ¯exibility of 2, both performance measures are enhanced under all conditions by providing more initial training. This is mainly due to the reduction in relearning losses, since these ¯exibility acquisition policies have been speci®cally designed to overcome the initial learning eects prior to implementing ¯exibility in the shop. More initial up-front training with FAP-1 and FAP-2 policies does not necessarily improve
b
a
3
2
3
2
2 4 8 12 16 20 40 2 4 8 12 16 20 40
2 4 8 12 16 20 40 2 4 8 12 16 20 40
1.6766 1.4525 1.2850 1.2128 1.1716 1.1447 1.0838 1.9148 1.6699 1.4571 1.3547 1.2929 1.2510 1.1514
1.9935 1.8032 1.6280 1.5335 1.4703 1.4237 1.2946 2.4297 2.3611 2.3132 2.2933 2.2820 2.2747 2.2582
100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
100.00 100.00 100.00 100.00 100.00 100.00 100.00 48.51 48.51 48.51 48.51 48.51 48.51 48.51 1.1986 1.1275 1.0737 1.0519 1.0402 1.0328 1.0172 1.2432 1.1733 1.1080 1.0785 1.0618 1.0511 1.0276
1.2271 1.1725 1.1250 1.1017 1.0871 1.0768 1.0503 1.3213 1.4179 1.5883 1.6998 1.7761 1.8316 1.9763 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
100.00 100.00 100.00 100.00 100.00 100.00 100.00 48.51 48.51 48.51 48.51 48.51 48.51 48.51
b
1.5308 1.3634 1.2328 1.1753 1.1421 1.1203 1.0703 1.6679 1.4937 1.3371 1.2615 1.2158 1.1850 1.1115
1.6541 1.5043 1.3631 1.2902 1.2443 1.2124 1.1329 1.8386 1.7570 1.6886 1.6567 1.6377 1.6249 1.5946
APT
100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
100.00 100.00 100.00 100.00 100.00 100.00 100.00 70.65 70.65 70.65 70.65 70.65 70.65 70.65
Final eciency
FAP-0 Final eciency
b
APT
FAP-1
b
Final eciency
FAP-0
APT
Batch size 25
Batch size 10 b
1.2907 1.1404 1.0847 1.0611 1.0481 1.0397 1.0215 1.2499 1.1838 1.1185 1.0878 1.0701 1.0585 1.0325
1.2289 1.1672 1.1108 1.0842 1.0685 1.0581 1.0339 1.2914 1.3031 1.3415 1.3717 1.3940 1.4108 1.4571
APT
FAP-1
100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
100.00 100.00 100.00 100.00 100.00 100.00 100.00 70.65 70.65 70.65 70.65 70.65 70.65 70.65
Final eciency
b
1.3005 1.2192 1.1477 1.1140 1.0939 1.0804 1.0484 1.3477 1.2707 1.1917 1.1513 1.1264 1.1092 1.0674
1.3186 1.2411 1.1675 1.1312 1.1091 1.0938 1.0574 1.3720 1.3152 1.2487 1.2104 1.1849 1.1666 1.1184
APT
FAP-0
100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
Final eciency
Batch size 250
Attrition interval (atr)/Training Time (tr) ratio is based on the training time (480 min) needed to fully learn in one department. Average processing time (APT) adjusted by the probability of attrition.
95%
85%
Forget- Flexi- atr/tr ting rate bility ratio a
Table 1 Results of the numerical analysis of the LFL model in a DRC for the FAP-0 and Fap-1 ¯exibility acquisition policies
1.2288 1.1679 1.1124 1.0861 1.0705 1.0601 1.0357 1.2603 1.2054 1.1451 1.1139 1.0946 1.0814 1.0495
1.2364 1.1779 1.1218 1.0944 1.0778 1.0666 1.0401 1.2728 1.2328 1.1839 1.1558 1.1373 1.1240 1.0897
b
FAP-1 APT
100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
Final eciency
168 H.V. Kher et al. / European Journal of Operational Research 115 (1999) 158±172
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169
Fig. 3. Average processing time as a function of attrition rate at FAP 0 ¯exibility acquisition policy.
performance with respect to a greater degree of ¯exibility. Acquiring ¯exibility of 3 is still costly when forgetting rate is high (FR 85%) and small
batches are used to transfer the worker between departments. For the combination of a forgetting rate of 85% and batch size of 10 jobs, even the
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provision of large initial training was not sucient to ensure complete learning of tasks (®nal eciency of 100%) across three departments. In conclusion, three important caveats emerge from this analysis. First, the extent of forgetting has an important eect on the performance of the worker during the cross-training period. Second, as the results in Table 1 indicate, acquiring ¯exibility appears to reduce the worker's ®nal eciency level in the harsh forgetting environments (FR 85%). Only the forgetting environment of 95% was shown to be conducive to learning three dierent tasks. Finally, the combination of ¯exibility acquisition policy and transfer policy (controlled by batch size) aects performance. The implications of these results is that when forgetting rate is 85%, the only way to eectively train the worker while reducing the learning and relearning related eciency losses is through the use of large batch sizes, which in turn reduces available ¯exibility. However large batch sizes are not needed when the worker is trained to perform tasks in only two dierent departments. Also, the use of extensive initial training helps considerably from an eciency point of view. The comparison of performance for the three ¯exibility acquisition policies in acquiring ¯exibility of 2 and 3 under conditions of 85% forgetting rate is shown in Fig. 4. Only the extreme cases with atr/tr ratios of 2 and 40 are shown. FAP-2 is seen to dominate the three policies as evidenced by the APT performance measure. Fig. 4 also shows how attrition rates and the transfer batch sizes aect the performance measures when ¯exibility is being acquired. Flexibility of 3 comes at a high cost (APT) when no initial training is provided (FAP-0) and small transfer batch sizes are used for transferring worker. The APT is also seen to increase with the attrition rate (atr/tr ratio). However, the combination of extensive up-front training and the use of large batch sizes appears to be the only solution to reduce APT when training the worker in three departments (F 3). These two conditions limit the bene®ts of higher ¯exibility, thereby suggesting that training workers in three departments is not a viable option when both signi®cant learning and relearning losses are present.
8. Conclusion One of the major contributions of the DRC system analysis reported in this study was on identifying the conditions under which a worker may not be able to learn even two dierent tasks. When coupled with 85% forgetting rate, the extent of learning and relearning losses under high attrition rate conditions (atr/tr ratios of 2 and 4) are such that they do not permit the worker to even acquire standard processing time eciency. Managers are much better o in such situations to focus their eorts more at increasing the retention rates of their workers rather than giving them additional training. It must be noted here that this result and the associated recommendation is at variance with much of the literature in DRC systems, which advocates that incremental ¯exibility is always useful (Treleven, 1989). However, prior literature has also not simultaneously modeled learning and forgetting phenomenon in reaching such a conclusion. The interaction between batch sizes and ¯exibility acquisition policy was another valuable ®nding of this study. In an eort to decrease manufacturing lead times and compete on the time based dimension, ®rms are moving towards small batch production (Stalk, 1988). Smaller batch sizes however have the detrimental eect of interrupting the learning process. Again, our ®ndings have identi®ed the conditions under which the ®rm would be better o in not training the workers in additional departments if retaining smaller lead times is an important dimension of being competitive in the market place. Future research studies should build on this work to further examine the ¯exibility ± learning loss tradeos under more dynamic conditions where the bene®ts associated with worker ¯exibility would be more readily apparent and measured. A more complex study that is perhaps based on the simulation methodology, and which models interaction eects between several workers rather than a single one, is necessary for this purpose. The impact of learning and forgetting losses can also be investigated on other inventory and customer service related performance
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Fig. 4. Average processing time as a function of ¯exibility acquisition policies.
measures that are dierent from the ones reported here. Finally, the ¯exibility gain-productivity loss tradeos must be further explored for
those ®rms where dierential rates of learning and forgetting exist across dierent departments of the shop.
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References Adam, E., Ebert, R.J., 1992. Production and Operations Management. Prentice-Hall, Englewood Clis, NJ. Adler, G.L., Nanda, R., 1974a. The eects of learning on optimal lot size determination ± single product case. AIIE Transactions 6, 14±20. Adler, G.L., Nanda, R., 1974b. The eects of learning on optimal lot size determination ± multiple product case. AIIE Transactions 6, 21±27. Anderlohr, G., 1969. What production breaks cost. Industrial Engineering, 34±36. Bailey, C.D., 1989. Forgetting and learning curve: A laboratory study. Management Science 35 (3), 340±352. Balo, N., 1966. Startups in machine intensive production systems. Journal of Industrial Engineering 17 (1), 25±32. Balo, N., 1970. Startup management. IEEE Transactions on Engineering Management EM-17, 132±141. Balo, N., 1971. Extension of the learning curve ± some empirical results. Operations Research Quarterly 22 (4), 329±340. Bobrowski, P.M., Park, P.S., 1993. An evaluation of labor assignment rules when worker are not perfectly interchangeable. Journal of Operations Management 11, 257±268. Carslon, J.C., Rowe, A.J., 1976. How much does forgetting cost? Industrial Engineering 8, 40±47. Crawford, J.R., 1944. Learning Curve, Ship Curve, Ratios, Related Data. Lockheed Aircraft Corporation. Fry, T.D., Kher, H.V., Malhotra, M.K., 1995a. Managing worker ¯exibility and attrition in dual resource constrained job shops. International Journal of Production Research 33 (8), 2163±2179. Fry, T.D., Steele, D.C., Saladin, B.A., 1995b. The role of management accounting in the development of a manufacturing strategy. International Journal of Operations and Production Management 15 (12), 21±31. Fryer, J.S., 1974. Labor ¯exibility in multiechelon dualconstraint job shops. Management Science 20 (7), 1073± 1080.
Globerson, S., 1980. The in¯uence of job related variables on the predictability power of three learning curve models. AIIE Transactions 12, 64±69. Globerson, S., Levin, N., Shtub, A., 1989. The impact of breaks on forgetting when performing a repetitive task. IIE Transactions 21 (4), 376±381. Hirsch, W.Z., 1956. Firm progress ratios. Econometrics 24 (2), 1245±1257. Homan, T.R., 1968. Eect of prior experience on learning curve parameters. Journal of Industrial Engineering 19, 412±413. Kertesz, L., 1987. Progressive work pacts split UAW; Lasting anger is feared. Automotive News, p. 1. Malhotra, M.K., Ritzman, L.P., 1990. Resource ¯exibility issues in multistage manufacturing. Decision Sciences 21 (4), 673±690. Malhotra, M.K., Fry, T.D., Kher, H.V., Donohue, J.M., 1993. The impact of learning and labor attrition on worker ¯exibility in dual resource constrained job shops. Decision Sciences 24 (3), 641±663. Rohan, T.M., 1987. Bosses ± who needs `em? Industry Week, 15±16. Schonberger, R.J., 1982. Japanese Manufacturing Techniques: Nine Hidden Lessons in Simplicity. Free Press, New York. Stalk, G., 1988. Time ± The next source of competitive advantage. Harvard Business Review 66, 41±51. Sule, D.R., 1978. The eect of alternate periods of learning and forgetting on economic manufacturing quantity. AIIE Transactions 10 (3), 338±343. Treleven, M., 1989. A review of dual resource constrained system research. IIE Transactions 21 (3), 279±287. Wisner, J.D., Pearson, J.N., 1993. An exploratory study of the eects of operator relearning in a dual resource constrained job shop. Production and Operations Management Journal 2 (1), 55±68. Yelle, L.E., 1979. The learning curve: Historical review and comprehensive survey. Decision Sciences 10, 302±328.
Theory and Methodology
Modeling simultaneous worker learning and forgetting in dual resource constrained systems Hemant V. Kher a, Manoj K. Malhotra b
b,*
, Patrick R. Philipoom b, Timothy D. Fry
b
a Division of Business and Economics, University of South Carolina Sumter, 200 MillerRoad, Sumter, SC 29150-2498, USA Department of Management Science, The Darla Moore School of Business, University of South Carolina, Columbia, SC 29208, USA
Received 1 July 1997; accepted 1 April 1998
Abstract This paper addresses issues related to modeling worker learning and forgetting eects in Dual Resource Constrained (DRC) systems. The learn-forget-learn (LFL) model of Carlson and Rowe (Carlson, J.C., Rowe, A.J., 1976. Industrial Engineering 8, 40±47) is used to critically evaluate several worker training related issues that are associated with the learning and forgetting phenomenon. A numerical analysis is performed on the LFL model within a DRC system to gain insights into the nature of relationships between the extent of worker ¯exibility, forgetting rates, attrition rates, and ¯exibility acquisition policies. Results suggest that in the presence of higher attrition and forgetting rates, a worker may not be able to achieve full eciency in as little as two dierent departments. Thus acquiring even incremental worker ¯exibility under such conditions may be infeasible. We also show that managers can use dierent ¯exibility acquisition policies designed in this study to reduce relearning losses and consequently improve the system performance. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Numerical analysis; Modeling; Scheduling; Dual-resource constrained systems
1. Introduction In response to an increased emphasis on lower inventories, on-time deliveries, and higher and consistent product quality, operations managers are striving to improve their manufacturing processes. One such method for process improvement is the development of ¯exible resources, in partic-
*
Corresponding author. Tel.: 001 803 777 2712; fax: 001 803 777 6876; e-mail: [email protected].
ular workforce ¯exibility (Malhotra and Ritzman, 1990). Flexible workers can provide a buer against uncertainties that are inherently present in any manufacturing system such as machine breakdowns, product mix changes, external demand changes, and material shortages. A ¯exible workforce also helps in reducing work-in-process inventory levels, thereby allowing for improvements in manufacturing lead times and customer service performance (Treleven, 1989). Japanese ®rms such as Mitsubishi Belting and Matsushita Electric are known to encourage worker ¯exibility
0377-2217/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 8 ) 0 0 1 9 0 - 8
H.V. Kher et al. / European Journal of Operational Research 115 (1999) 158±172
by tying wages directly to the dierent set of skills possessed by the workers (Schonberger, 1982). Similarly, studies by Rohan (1987) and Kertesz (1987) indicate that ®rms like Frito Lay and General Motors have adopted worker ¯exibility to gain from its bene®ts. Though bene®ts associated with a ¯exible crosstrained workforce are well understood, the same cannot be said for the cost of acquiring and utilizing worker ¯exibility. The costs occur in part during the cross-training period due to worker learning eects, which result in productivity losses since the workers are typically less ecient during training. The losses during the learning phase can be further exacerbated if interruptions occur due to varying workloads between departments causing workers to leave an under loaded department before training is complete. The ongoing use of worker ¯exibility after complete training has occurred may also be expensive if prolonged absence from a given department leads to worker forgetting. Similarly, attrition can also add to learning losses if the new worker needs training in order to be utilized ¯exibly. Thus acquiring worker ¯exibility can result in a one time training loss, while forgetting and relearning represent the ongoing cost of using it. Bobrowski and Park (1993) give speci®c examples of ®rms incurring training related losses (Chrysler corporation used 900,000 h before launching the new Jeep, and GM provided each worker over 80 h of training before starting new product lines (p. 258)). Similarly, Wisner and Pearson (1993) report that ®ve of the six DRC shop managers they surveyed for their study attested to the presence of worker forgetting when transfers took place, and that three of these managers used frequent periodic training to control their relearning problem. The preceding discussion suggests that both learning and forgetting (which leads to relearning) phenomenon must be modeled and studied simultaneously in order to truly understand how worker ¯exibility should be acquired and used to provide ongoing bene®ts to the ®rm. Such an understanding becomes especially critical for a large number of ®rms that are known to rely on labor
159
productivity as a primary measure of shop performance (Fry et al., 1995b). In such ®rms, the productivity losses attributable to worker learning and relearning may be a signi®cant source of concern for measuring performance and setting standards. In this paper, we perform numerical analysis on a well known worker learning and forgetting model proposed by Carslon and Rowe (1976). This is done in the context of understanding several training related issues in the management of dual resource constrained (DRC) systems. Given the detrimental eect that productivity losses can have on inventory and customer service performance of a DRC ®rm, managers must carefully plan the process of acquiring and operationalizing ¯exibility in the presence of learning and relearning eects. Important issues that these managers must address include deciding the number of dierent tasks for which the workers should be trained (termed as the degree of ¯exibility), how to train the workers, and how to assign workers to dierent tasks during the crosstraining period so as to maximize learning and minimize the relearning losses. Addressing these training and worker assignment issues in the simultaneous presence of learning and relearning will facilitate a better understanding of how many dierent tasks a worker can eectively learn in dierent types of learning and forgetting environments. A review of the literature on the modeling of learning, forgetting, and relearning is presented in Section 2 to justify the choice of the learn-forgetlearn (LFL) model that is used to capture these eects. Managerial policies related to the ¯exibility acquisition process are discussed in Section 3. A brief description of the one worker DRC system which serves as the vehicle for the numerical analysis of the LFL model can be found in Section 4. The performance measures associated with the numerical analysis are discussed in Section 5. We discuss the details of the LFL model as implemented in our DRC system in Section 6. Results of the analysis are presented in Section 7, and the conclusions of the study are ®nally presented in Section 8.
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2. Modeling learning and relearning The consideration of learning eects when setting labor standards has been well documented. Even though several learning models have been oered over time Yelle (1979), the log-linear model of Crawford (1944) has been used most frequently in a variety of studies to capture the learning effects. In this model, unit processing times improve at a uniform rate as the cumulative number of units produced by the learning workers is doubled. The rate at which workers learn, the time required to fully learn new tasks, as well as the associated labor productivity losses in a DRC environment are a function of several dierent factors. These factors include the nature of task being performed (e.g. machine or labor intensive tasks (Hirsch, 1956), planning of the training programs, training facilities, monetary incentives oered to the learning workers during the training period, etc. (Balo, 1970, 1971; Yelle, 1979)). As these factors are not constant across dierent industries, the learning rates, the length of the training period, and the labor productivity losses can be expected to vary. In situations where fully trained workers reach an ``ideal'' standard processing time and no further improvement is possible, a plateau log-linear learning model is more appropriate. Such a condition would readily exist when machine capability ultimately determines unit processing time, and has been observed in industry (Balo, 1966; Yelle, 1979). The plateau log-linear model has been previously used in DRC studies (Malhotra et al., 1993), and is shown as ÿ ÿL Yn1 Max U1 Xn1 ; 1
1
where Yn1 is the time required to process the next job, U1 the initial processing time factor, L a constant determined by log10 (learning rate)/ log10 (0.5) and Xn1 is the next job to be processed. In contrast to learning, the literature on the modeling of relearning is not extensive, although several studies have acknowledged its existence in practical settings (Bailey, 1989; Globerson, 1980; Globerson et al., 1989). Bailey (1989) investigated the relearning phenomenon in a laboratory setting,
and using a single break in the learning process showed that forgetting of a task depends on the nature of tasks being performed (continuous control versus procedural tasks), forgetting rate, amount learned prior to the interruption, and the length of the interruption interval. Similarly, studies by Globerson (1980) and Globerson et al., (1989) also used a single break in the learning process, and indicated that the power curves similar to that shown in Eq. (1) provide the ``best ®t'' in modeling relearning. These studies also showed that learning after interruption followed the traditional learning pattern, and that the initial processing time for the ®rst job to be processed after the interruption was a function of the length of the interruption interval as well as prior experience. Forgetting eects have been estimated in the existing literature by using both subjective guesses (Anderlohr, 1969), as well as mathematical models (Homan, 1968). Among the available approaches, the learn-forget-learn (LFL) model proposed by Carslon and Rowe (1976) has been used to capture forgetting eects in several dierent lot sizing studies (e.g. Adler and Nanda, 1974a, b; Sule, 1978). This log-linear model assumes that when an interruption occurs during learning, the worker will move back up on the learning curve. The exact point to which experience deteriorates depends on the amount of experience gained prior to the interruption, the forgetting rate, and the length of the interruption interval. The extent of forgetting is thus captured using a power model, and is consistent with the ®ndings of Globerson et al. (1989). The LFL model determines the processing time for the n 1th job in accordance with Eq. (1). However, the resumption point after every interruption is calculated as h i ÿF =L 1; 2 Xn1 Xn 1F =L Xn e where Xn1 is the ®rst job to be processed after the interruption interval, Xn the number of jobs processed thus far, F is a constant determined by log10 (forgetting rate)/ log10 (0.5) and e is the number of jobs that could have been processed during the interruption interval. In Eq. (2), the term e is calculated by determining the eciency of the worker at the point of
H.V. Kher et al. / European Journal of Operational Research 115 (1999) 158±172
interruption. Speci®cally, if the interruption occurs when the worker is processing jobs at the ideal standard rate (i.e. with complete eciency), then the number of jobs that could have been processed by the worker during the interruption interval is computed by simply dividing the interruption interval by the standard processing time. However, if the worker is interrupted during the learning period, then the actual number of jobs that could have been processed during the interruption interval is determined as if learning had continued during the interruption interval. Thus the LFL approach takes into account the exact level of experience gained by the worker just before interruption occurs. In contrast, Wisner and Pearson (1993) modeled the relearning phenomenon very dierently from the prior research and the LFL model described above. Relearning rates of 0%, 30%, and 60% were studied, where a 30% relearning implied that the ®rst job processed by a worker after a transfer has a processing time of 130% of the standard processing time. Further, as the consecutive jobs processed at a machine doubled, the processing time decreased at a rate of 70%, thereby implying a 70% rate of learning after the interruption. Thus their model diers from the LFL model in two important ways. First, the amount of prior experience possessed by that worker is ignored. Every interruption is treated the same, irrespective of the workers prior experience. Secondly, the length of the interruption interval is also ignored in determining the extent of relearning losses. Our own experiences with actual industrial situations indicate that due to all the factors considered, the LFL model more accurately depicts the forgetting and relearning process in practical settings than the Wisner and Pearson (1993) model. In particular, we cite the situation at a plant that manufactures automatic switching systems. The learning takes place with respect to elaborate setups at machines. Sometimes, these setups can well exceed the actual processing time of the job that uses that setup. Workers can take as long as six months to become pro®cient at a machine, and then forget that skill when transferred to another department in the shop. Return to the
161
same department does not start at full pro®ciency, but is dependent upon how long the operator had spent at that machine in the past before the transfer occurred. The forgetting and relearning still takes place even after multiple breaks of this nature, provided the task itself is of sucient complexity and takes a long time to master in the ®rst instance. How best to train and ¯exibly deploy the workers in such ®rms is not known, since the LFL model has not been previously examined in dual resource constrained settings. The best policies may lie in minimizing the loss of eciency that will inevitably result when a worker learns and forgets too much due to a transfer to another department. The degree of learning loss will depend upon the length of absence, and transferring a worker may not even be worthwhile if relearning will severely diminish worker productivity. Yet, not transferring a worker at all forgoes the well documented bene®ts associated with worker ¯exibility (e.g. Fryer, 1974b; Treleven, 1989; Malhotra et al., 1993). These trade-os, which primarily spring from the policies used to train and deploy workers within the shop, must be better understood in order to further our knowledge of worker ¯exibility in DRC systems. 3. Managerial policies and operating environment The best set of policies that can be used to train workers to acquire the desired level of ¯exibility in DRC systems, as well as the worker assignment policies that can be used to reduce the relearning eects, are not immediately apparent. We propose that managers can acquire the desired level of ¯exibility through dierent ¯exibility acquisition policies (or training schemes). We will de®ne such policies here, and test them by conducting a numerical analysis of the LFL model across a wide range of conditions that characterize the operating environment of a DRC system. The relevant operating factors would include the attrition rate of workers and level of ¯exibility acquired. It could be argued that giving more training to workers under conditions of high learning and forgetting losses may not be worthwhile if they are going to
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leave the system due to attrition before the bene®ts of their training have been fully realized. 3.1. Worker attrition Worker attrition directly impacts the performance of DRC shops since newly hired workers require greater training than workers who have been in the shop for a period of time. In other studies (Fry et al., 1995a; Malhotra et al., 1993), worker attrition has been modeled simply as a proportion of workers lost per year. Modeling attrition in this fashion does not capture its true impact, since even a small rate of attrition may be signi®cant if the time required to train a worker is relatively long. In order to better capture the relationship between attrition rate and the time required for training, we use the ratio between attrition interval and the time required for training. Here the attrition interval is simply the mean time between consecutive attritions. Recognizing this relationship represents a major improvement in the study of worker ¯exibility in DRC shops. We model seven dierent worker attrition interval to training period ratios (atr/tr) of 2, 4, 8, 12, 16, 20, and 40, respectively. A ratio of 2 implies that the worker attrition interval is twice as large as the training period and represents the harshest setting, while larger ratios represent relatively milder attrition environments. These attrition interval ratios can also be converted into an average annual attrition rate. Consider for example a DRC system which has 12 workers. If it takes 4 weeks to train a worker, then an atr/tr ratio of 2 implies that the shop will lose 6.25 workers per year (50 weeks), thereby implying a 52.08% annual attrition rate. Similarly with ratios of 4, 8, 12, 16, 20 and 40, the corresponding annual attrition rates in the above example would be 26.04%, 13.02%, 8.68%, 6.51% 5.20%, and 2.60%, respectively. 3.2. Flexibility acquisition policies As used in this study, the ¯exibility acquisition policy represents the manner in which workers are trained. One such policy is to move workers after
each job is completed, thereby training workers simultaneously in several dierent departments. Another widely divergent policy may be to train workers in only one department for an extended period of time before allowing them to begin training in a new department. These two policies represent the two extremes between the desire for greater ¯exibility right away and the desire for greater worker eciency or productivity over the long run. We introduce three ¯exibility acquisition policies (FAP) to represent this tradeo. The ®rst level (FAP-0) involves training a worker to perform tasks in the desired number of departments with no restrictions placed on the transferring worker. The worker is transferred to the next eligible department to perform a task after processing a given batch size, and before complete learning has taken place in the current department. This approach allows the DRC system to better utilize worker ¯exibility during the cross-training period by frequent transfers, but would also likely increase the relearning eects and decrease worker productivity. At the second level (FAP-1), the worker is not allowed to transfer until the ideal standard is reached, while FAP-2 requires the worker to train in one department for twice the number of jobs required to reach the ideal standard. FAP-2 thus represents a conservative policy that fosters a greater level of worker eciency. It could be used when the management does not want to incur large relearning losses during the initial learning periods when ¯exibility is being acquired. 3.3. Worker transfer policies The worker training policies mentioned above control only the timing of the very ®rst transfer for a worker from a given department. Once the desired level of training has been provided in that department, transfer policies are used to determine ``when'' and ``where'' to transfer the worker. The ``where'' rule has been shown to be a less important determinant of performance than the ``when'' rule. Hence it is ®xed in this study by sequentially alternating a worker in the eligible departments of
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163
the DRC system modeled. The ``when'' rule has been shown to have a signi®cant impact on the DRC shop performance by a large number of studies (see e.g. Treleven, 1989). It essentially de®nes the frequency of worker transfers. More frequent transfers lead to a more ¯exible use of the workforce, but also increase the number of interruptions in the learning process, thereby reducing worker eciency due to an increase in the relearning losses. In our DRC system, the ``when'' rule is implemented by using the concept of batch size. The batch size is also the parameter used in transferring the worker between departments during the FAP-0 training scheme. As an experimental factor, batch size is set at three dierent levels starting with a low of 10 jobs to a high of 250 jobs. At the low end, the worker would spend at least 10 time units in a department (after the training scheme has been implemented), and perhaps longer if the ideal standard has not been reached. Given the transfer time penalties associated with forgetting, it was not considered reasonable to process less than 10 (or more than 250) jobs in a given department. In general, smaller batch sizes will reduce the worker's stay in any department and also lead to more interruptions in the learning process. On the other hand, while larger batch sizes appear to restrict bene®ts of ¯exibility, they will also reduce relearning losses. The dierences between dierent ¯exibility acquisition policies will also diminish with the use of larger batch sizes, since the batch policy and ¯exibility acquisition policy will interact with each other during the training phase of the workers. Once complete training has been achieved, batch sizes will determine the actual amount of ¯exibility utilized.
¯exibility of two and three respectively, and were chosen from a prior study which suggests that training workers to perform tasks in more than three dierent departments in the presence of high learning losses can considerably worsen system performance (Malhotra et al., 1993). Learning and relearning eects during the cross-training period are captured for the worker by using the LFL model (Eq. (2)). Based on prior literature (Malhotra et al., 1993), an 85% learning rate is selected with the initial unit processing time in a department being 400% of the standard time. These parameter values have been shown in this earlier study to represent challenging learning environments with complex tasks. We are less interested here in modeling simpler learning situations where the forgetting and relearning phenomenon is not signi®cant enough to warrant management's attention. Consequently, we also choose two different forgetting rates of 85% and 95% to represent a reasonably challenging and wide range of forgetting conditions. Finally, the worker's learning and forgetting are not department speci®c. The numerical analysis proceeds by training the worker to process jobs in two or three dierent departments, depending upon the degree of ¯exibility being investigated. Each department is assumed to have a sucient number of jobs waiting to be processed such that it is never starved for work. The worker is sequentially assigned to different departments after processing the given batch size of jobs in each department. The total number of cycles executed by the worker depend upon worker attrition, which is modeled using the exponential distribution. The attrition interval which was described in the preceding section serves as the mean value for this distribution. The probability of attrition is computed at the completion of each cycle using>
4. The DRC system
CPn
Worker training and deployment related issues are investigated by performing a numerical analysis of the LFL model in the context of a single worker in a DRC system who is trained to process jobs in two or three functionally dierent departments (or machines). These represent levels of
n X P atri ; i1
P atri eÿFTi =INT ÿ eÿSTi =INT
8 i 1; . . . ; n; 3
where CPn is the cumulative probability of worker attrition over the ®rst n cycles P(atr)i the probability of attrition during the ith cycle, STi the start
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time of the ith cycle, FTi the completion time of the ith cycle and INT is the length of the attrition interval (mean attrition inter-arrival time). The cycle is chosen as a unit of analysis since it permits the division of the attrition interval into smaller discrete portions. The probability of worker leaving within each cycle is computed based upon the selected attrition rate, and added to obtain the cumulative probability of attrition. Analysis of performance for a given set of conditions is terminated when this cumulative probability of worker attrition equals unity.
processing time of one unit by the actual processing time of the worker at the end of the analysis. This de®nition of labor eciency is also consistent with that used in prior literature (e.g. Adam and Ebert, 1992, p. 12). If the forgetting rates are high, it is quite possible that even though the system has reached steady state, a worker may not become fully ecient before leaving the system. In that case, the worker's steady state processing time at the end of the analysis will be greater than the standard processing time of one time unit, and the ®nal level of eciency will be less than 100%.
5. Performance measures
6. Implementation of the model
The objective behind the analysis is to understand the learning and forgetting process in the presence of attrition, and how ¯exibility acquisition and worker assignment policies aect performance when managers attempt to attain dierent levels of worker ¯exibility. As such, we use two criteria to assess these impacts. These two criteria are the average job processing time for the tenure of employment for the worker (APT), and the ®nal eciency of the worker. The APT measure is computed as
The steps involved in performing a numerical analysis of the LFL model in a DRC setting for a single worker are explained here. The DRC system where the worker is trained to process jobs in two or three functionally dierent departments is illustrated in Fig. 1. In conducting the numerical analysis, it is necessary to determine the batch processing time (BPT) in a given department for a given batch size. In addition, it is also necessary to determine the quantity e, which represents the number of jobs that could have been processed in the interruption interval if the learning were to continue. The quantity e is then used in Eq. (2) to determine the resumption point from where learning is to resume for the worker in a given department. Note that since the forgetting rate (F) is a continuous variable, the cumulative number of jobs completed is not always an integer quantity. Consequently, the starting and ending job numbers (which represent the batch size that is processed) over which BPT is to be determined is also not guaranteed to always be an integer quantity. Similarly, the quantity e also may not always be an integer number in general. Fig. 2 is useful in understanding the equations used to compute the terms BPT and e. Case 1 represents the situation where the processing of a batch begins and ends on the slope of the learning curve. Case 2 represents the situation where the processing of a batch begins on the slope, but ends on the plateau region of the curve. Finally, Case 3
APT
n X i1
P atri FTi ÿ STi =Ni n;
4
where APT is the attrition adjusted average processing time over all n cycles, Ni the number of jobs processed during the cycle i and n is the total number of cycles. The APT measure is computed at the end of each cycle by multiplying the average processing time for each job during that cycle (length of the cycle/number of jobs processed during the cycle) by the probability of attrition during the cycle. When the cumulative probability of worker attrition approaches unity, the sum of attrition adjusted processing times is divided by the total number of cycles to determine the attrition adjusted average processing time (APT). The second criterion used to measure performance is the ®nal level of eciency achieved by the worker. It is computed by dividing the standard
H.V. Kher et al. / European Journal of Operational Research 115 (1999) 158±172
165
Fig. 1. Schematic of the DRC shop for numerical analysis.
represents the situation where the processing of a batch begins and ends on the plateau. Expressions for computing BPT for each of the three cases are derived as shown below. It must be noted here that BPT is the dierence (FTi ) STi ) that was used in Eq. (4) to compute the APT. Case 1: Slope to slope XZ BS
U1 X
BPT h
ÿL
Case 2: Slope to plateau. This expression is derived by dividing the integral into the slope and plateau regions Zb
XZ BS
dx
X
b
BS ÿ b ÿ X ;
X
5
where X is the very ®rst job number in the current batch, BS the batch size and other terms (U1 and L) are de®ned as before.
1 dx
h i ÿL1 ÿL1 fU1 = 1 ÿ Lg b 1 ÿ X 1
dx fU1 = 1 ÿ Lg
i X BS 1ÿL1 ÿ X 1ÿL1 ;
U1 X
BPT
ÿL
6
where b is the intersection of the slope and the plateau regions of the learning curve. Case 3: Plateau to plateau XZ BS
1 dx BS:
BPT X
7
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H.V. Kher et al. / European Journal of Operational Research 115 (1999) 158±172
Fig. 2. Illustration of Cases 1, 2, and 3 for batch processing time and interruption interval calculations.
Expressions for computing e, the number of jobs that could have been processed during the interruption interval if learning were to continue, are shown below. Case 1: Slope to slope h i1ÿL ÿL1 ÿ X ÿ 1; e fIT 1 ÿ L=U1 g X 1 8 where IT is the length of the interruption interval expressed in time units. Case 2: Slope to plateau h i ÿL1 ÿL1 ÿ X 1 e fU1 = 1 ÿ Lg b 1 b ÿ X:
9
Case 3: Plateau to plateau e IT:
10
In order to better understand the implementaion of our DRC system, consider an example situation where the worker is being trained to process jobs in two dierent departments. In this case, a cycle is completed when the worker sequentially
processes the predetermined batch size in each of the two departments (i.e. department #1, and department #2). For the very ®rst visit in a given department, the batch size to be processed is the maximum of the quantity speci®ed by the ¯exibility acquisition policy or that used for transferring the worker. Thus in the setting where FAP-1 ¯exibility acquisition policy is used and the worker is transferred using a batch size of 250, the worker must process 370 units at the very ®rst visit in each of the two departments. If the FAP-0 policy were to be used rather than FAP-1 in the above example, then the worker would be required to process 250 jobs in each of the two departments. For the subsequent cycles, the worker is required to process the number of jobs speci®ed by the batch size which is used to make the transfer decision (i.e. 250 jobs in the above example). The speci®c point from where the worker resumes learning in a given department is calculated using Eq. (2). The value of e in Eq. (2) is given either by Eq. (8), or by Eq. (9), or by Eq. (10) depending upon the exact case under consideration. The existence of any case scenario 1±3 depends upon the worker's prior
H.V. Kher et al. / European Journal of Operational Research 115 (1999) 158±172
experience in the department, the length of the interruption interval, and the forgetting rate. The two performance measures are collected at the end of ®nal cycle for the worker, whereby the total number of cycles (n) executed by the worker are dependent upon the existing worker attrition rate. The ®nal eciency of the worker is computed when the worker departs the DRC system. The second performance measure of APT for any given policy (or experimental cell) under consideration is obtained by using Eq. (4), with the requisite expression for BPT being given by Eqs. (5)±(7). This model was implemented through a Fortran program written on a UNIX based IBM RISC system. The model code was validated by matching its results with calculator based computations of the relevant equations carried out for small size example problems. In all cases examined, the numerical results obtained through the two methods were identical. This allowed us to execute the experimental design described earlier. A single observation of the two performance measures for each experimental cell was obtained. No statistical analysis of the data was needed since the speci®cation of the LFL model used does not include any random variations. These results are discussed in the next section.
7. Results Table 1 shows the results for dierent performance measures for the FAP-0 and FAP-1 policies, while the graphical illustration of these results can be seen in Fig. 3. Based on Eq. (1), a worker reached the ideal standard for the 85% learning environment after processing 370 jobs consecutively in a given department for the FAP1 policy. In contrast, the worker in the FAP-2 policy was not allowed to transfer until 740 consecutive jobs had ®rst been processed in the department in which learning was taking place. The trend of the results for the FAP-2 policy were similar to the ones for FAP-1 policy, even though the average eciency of the worker during the learning period increased as we progressed towards the FAP-2 policy. Hence the results for
167
FAP-2 policy are not shown here in Table 1. However, we do subsequently identify the conditions under which managers should use FAP-2 acquisition policy to exploit the bene®ts of a trained worker. As can be seen in Fig. 1 for the FAP-0 policy, the APT for the incremental ¯exibility case (F 2) is always smaller than the APT for F 3. However, the dierences between the APT for these two ¯exibility levels decrease with a reduced forgetting rate (FR 95%). Table 1 shows that except for the highest attrition rates (atr/tr ratios of 2 and 4) and forgetting rate of 85%, a worker can be fully trained in two dierent departments (F 2) under all the other conditions tested as evidenced by ®nal eciency. In contrast, complete pro®ciency for ¯exibility of 3 can only be reached for high forgetting rates (85%) with very large batch sizes. Table 1 also shows that a worker's level of experience increases with increase in batch size. This in turn implies that as the batch size is increased, the worker is able to become more ecient as measured by APT and ®nal eciency. As expected, a reduction in forgetting rate (going from FR 85% to FR 95%) has a bene®cial impact on the worker performance. For a ®xed batch size and atr/tr ratio, the APT of the worker under the 95% forgetting environment is always better than the performance under the 85% forgetting rate (see Table 1). Performance also improves with increase in the atr/tr ratio, which causes the worker to stay in the shop for a longer period of time. This in turn allows him to become more ecient and improve the APT. Providing more training on one machine before transferring, as controlled by FAP-1 and FAP-2 policies improves performance. Results show that when acquiring ¯exibility of 2, both performance measures are enhanced under all conditions by providing more initial training. This is mainly due to the reduction in relearning losses, since these ¯exibility acquisition policies have been speci®cally designed to overcome the initial learning eects prior to implementing ¯exibility in the shop. More initial up-front training with FAP-1 and FAP-2 policies does not necessarily improve
b
a
3
2
3
2
2 4 8 12 16 20 40 2 4 8 12 16 20 40
2 4 8 12 16 20 40 2 4 8 12 16 20 40
1.6766 1.4525 1.2850 1.2128 1.1716 1.1447 1.0838 1.9148 1.6699 1.4571 1.3547 1.2929 1.2510 1.1514
1.9935 1.8032 1.6280 1.5335 1.4703 1.4237 1.2946 2.4297 2.3611 2.3132 2.2933 2.2820 2.2747 2.2582
100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
100.00 100.00 100.00 100.00 100.00 100.00 100.00 48.51 48.51 48.51 48.51 48.51 48.51 48.51 1.1986 1.1275 1.0737 1.0519 1.0402 1.0328 1.0172 1.2432 1.1733 1.1080 1.0785 1.0618 1.0511 1.0276
1.2271 1.1725 1.1250 1.1017 1.0871 1.0768 1.0503 1.3213 1.4179 1.5883 1.6998 1.7761 1.8316 1.9763 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
100.00 100.00 100.00 100.00 100.00 100.00 100.00 48.51 48.51 48.51 48.51 48.51 48.51 48.51
b
1.5308 1.3634 1.2328 1.1753 1.1421 1.1203 1.0703 1.6679 1.4937 1.3371 1.2615 1.2158 1.1850 1.1115
1.6541 1.5043 1.3631 1.2902 1.2443 1.2124 1.1329 1.8386 1.7570 1.6886 1.6567 1.6377 1.6249 1.5946
APT
100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
100.00 100.00 100.00 100.00 100.00 100.00 100.00 70.65 70.65 70.65 70.65 70.65 70.65 70.65
Final eciency
FAP-0 Final eciency
b
APT
FAP-1
b
Final eciency
FAP-0
APT
Batch size 25
Batch size 10 b
1.2907 1.1404 1.0847 1.0611 1.0481 1.0397 1.0215 1.2499 1.1838 1.1185 1.0878 1.0701 1.0585 1.0325
1.2289 1.1672 1.1108 1.0842 1.0685 1.0581 1.0339 1.2914 1.3031 1.3415 1.3717 1.3940 1.4108 1.4571
APT
FAP-1
100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
100.00 100.00 100.00 100.00 100.00 100.00 100.00 70.65 70.65 70.65 70.65 70.65 70.65 70.65
Final eciency
b
1.3005 1.2192 1.1477 1.1140 1.0939 1.0804 1.0484 1.3477 1.2707 1.1917 1.1513 1.1264 1.1092 1.0674
1.3186 1.2411 1.1675 1.1312 1.1091 1.0938 1.0574 1.3720 1.3152 1.2487 1.2104 1.1849 1.1666 1.1184
APT
FAP-0
100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
Final eciency
Batch size 250
Attrition interval (atr)/Training Time (tr) ratio is based on the training time (480 min) needed to fully learn in one department. Average processing time (APT) adjusted by the probability of attrition.
95%
85%
Forget- Flexi- atr/tr ting rate bility ratio a
Table 1 Results of the numerical analysis of the LFL model in a DRC for the FAP-0 and Fap-1 ¯exibility acquisition policies
1.2288 1.1679 1.1124 1.0861 1.0705 1.0601 1.0357 1.2603 1.2054 1.1451 1.1139 1.0946 1.0814 1.0495
1.2364 1.1779 1.1218 1.0944 1.0778 1.0666 1.0401 1.2728 1.2328 1.1839 1.1558 1.1373 1.1240 1.0897
b
FAP-1 APT
100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
Final eciency
168 H.V. Kher et al. / European Journal of Operational Research 115 (1999) 158±172
H.V. Kher et al. / European Journal of Operational Research 115 (1999) 158±172
169
Fig. 3. Average processing time as a function of attrition rate at FAP 0 ¯exibility acquisition policy.
performance with respect to a greater degree of ¯exibility. Acquiring ¯exibility of 3 is still costly when forgetting rate is high (FR 85%) and small
batches are used to transfer the worker between departments. For the combination of a forgetting rate of 85% and batch size of 10 jobs, even the
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H.V. Kher et al. / European Journal of Operational Research 115 (1999) 158±172
provision of large initial training was not sucient to ensure complete learning of tasks (®nal eciency of 100%) across three departments. In conclusion, three important caveats emerge from this analysis. First, the extent of forgetting has an important eect on the performance of the worker during the cross-training period. Second, as the results in Table 1 indicate, acquiring ¯exibility appears to reduce the worker's ®nal eciency level in the harsh forgetting environments (FR 85%). Only the forgetting environment of 95% was shown to be conducive to learning three dierent tasks. Finally, the combination of ¯exibility acquisition policy and transfer policy (controlled by batch size) aects performance. The implications of these results is that when forgetting rate is 85%, the only way to eectively train the worker while reducing the learning and relearning related eciency losses is through the use of large batch sizes, which in turn reduces available ¯exibility. However large batch sizes are not needed when the worker is trained to perform tasks in only two dierent departments. Also, the use of extensive initial training helps considerably from an eciency point of view. The comparison of performance for the three ¯exibility acquisition policies in acquiring ¯exibility of 2 and 3 under conditions of 85% forgetting rate is shown in Fig. 4. Only the extreme cases with atr/tr ratios of 2 and 40 are shown. FAP-2 is seen to dominate the three policies as evidenced by the APT performance measure. Fig. 4 also shows how attrition rates and the transfer batch sizes aect the performance measures when ¯exibility is being acquired. Flexibility of 3 comes at a high cost (APT) when no initial training is provided (FAP-0) and small transfer batch sizes are used for transferring worker. The APT is also seen to increase with the attrition rate (atr/tr ratio). However, the combination of extensive up-front training and the use of large batch sizes appears to be the only solution to reduce APT when training the worker in three departments (F 3). These two conditions limit the bene®ts of higher ¯exibility, thereby suggesting that training workers in three departments is not a viable option when both signi®cant learning and relearning losses are present.
8. Conclusion One of the major contributions of the DRC system analysis reported in this study was on identifying the conditions under which a worker may not be able to learn even two dierent tasks. When coupled with 85% forgetting rate, the extent of learning and relearning losses under high attrition rate conditions (atr/tr ratios of 2 and 4) are such that they do not permit the worker to even acquire standard processing time eciency. Managers are much better o in such situations to focus their eorts more at increasing the retention rates of their workers rather than giving them additional training. It must be noted here that this result and the associated recommendation is at variance with much of the literature in DRC systems, which advocates that incremental ¯exibility is always useful (Treleven, 1989). However, prior literature has also not simultaneously modeled learning and forgetting phenomenon in reaching such a conclusion. The interaction between batch sizes and ¯exibility acquisition policy was another valuable ®nding of this study. In an eort to decrease manufacturing lead times and compete on the time based dimension, ®rms are moving towards small batch production (Stalk, 1988). Smaller batch sizes however have the detrimental eect of interrupting the learning process. Again, our ®ndings have identi®ed the conditions under which the ®rm would be better o in not training the workers in additional departments if retaining smaller lead times is an important dimension of being competitive in the market place. Future research studies should build on this work to further examine the ¯exibility ± learning loss tradeos under more dynamic conditions where the bene®ts associated with worker ¯exibility would be more readily apparent and measured. A more complex study that is perhaps based on the simulation methodology, and which models interaction eects between several workers rather than a single one, is necessary for this purpose. The impact of learning and forgetting losses can also be investigated on other inventory and customer service related performance
H.V. Kher et al. / European Journal of Operational Research 115 (1999) 158±172
171
Fig. 4. Average processing time as a function of ¯exibility acquisition policies.
measures that are dierent from the ones reported here. Finally, the ¯exibility gain-productivity loss tradeos must be further explored for
those ®rms where dierential rates of learning and forgetting exist across dierent departments of the shop.
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