Production, Manufacturing and Logistics
Steady-state skill levels of workers in learning and forgetting environments: A dynamical system analysis

https://doi.org/10.1016/j.ejor.2013.06.009Get rights and content

Highlights

  • We study worker skill-levels’ long-term characteristics under learning and forgetting.

  • Our convergence results are applicable to general learning and forgetting functions.

  • Using the dynamical system analysis approach has given our paper theoretical rigor.

  • We provide numerous examples to support the practicality of our theoretical results.

Abstract

This article presents a study on the long-term (i.e., steady-state, convergence) characteristics of workers’ skill levels under learning and forgetting in processing units in a manufacturing environment, in which products are produced in batches. Assuming that all workers already have the basic knowledge to execute the jobs, workers learn (accumulate their skill) while producing units within a batch, forget during interruptions in production, and relearn when production resumes. The convergence properties in the paper are examined under assumptions of an infinite time horizon, a constant demand rate, and a fixed lot size. Our work extends the steady-state results of Teyarachakul, Chand, and Ward (2008) to the learning and forgetting functions that belong to a large class of functions possessing some differentiability conditions. We also discuss circumstances of manufacturing environments where our results would provide useful managerial information and other potential applications.

Introduction

This article investigates the problem of finding the long-term (i.e., steady-state) characteristics of batch production time in an environment where workers have to produce a fixed number of units of an item in batches at different points in time. Assuming that the workers already have the basic knowledge of operations to do the job, workers learn further while producing units within a batch, forget during interruptions in production, and relearn when production resumes; hence, learning and forgetting in processing time occurs intermittently in a batch manufacturing environment. We propose a model that will identify mathematical properties on general learning and forgetting functions that will be instrumental in studying the steady-state skill level of workers in such a batch manufacturing environment.

Under batch manufacturing, the equipment is shared by several products, and the products are produced in batches (lots). Therefore, workers have to go through the process of a learn-forget-relearn cycle repeatedly. During the periods of production and non-production of an item, respectively, workers’ skills (in producing that item) improve and decline, as measured by the instantaneous amount of time it takes them to complete a unit. Batch production time is defined as the amount of time to complete a production of q units within a batch. In order to analyze the steady-state characteristics, an infinite-horizon assumption is assumed. We also assume a constant demand rate d and a fixed lot size q.

In a learn-forget-relearn cycle manufacturing environment, scheduling tasks are complicated. Even for those tasks being performed by the same worker, depending on the worker’s skill level at the start of production, the batch production times for different production cycles could be inconsistent. Consequently, our steady-state results can help managers assess the long-term impacts on scheduling, costs, and time when producing the same item at the regular interval q/d. For instance, our steady-state batch production time can be used in computing the corresponding batch production-cost (such as labor cost); it is an important element of the average cost per cycle, in addition to inventory holding and setup costs. A manager can, in turn, use this information in determining the order quantity that is most cost effective (Teyarachakul et al., 2008). As to setting up standard times, Baloff (1966) suggested to use per-unit time when workers’ learning reach plateau. However, under the presence of learning and forgetting in batch manufacturing environments, the converged workers’ skill levels and the corresponding production times are particularly useful information in determining the appropriate standard times.

Accurate understanding of the long-term influence of learning and forgetting in production will aid in estimating productivity, determining economic lot size, setting time standards, estimating wage incentives, and predicting labor requirements (Nembhard & Osothsilp, 2001). For example, the steady-state production rate for producing a particular product is obtained as the ratio of lot size to the steady-state batch production time. This is valuable information for a manager seeking better long-term productivity; if the steady-state production rate is lower than required, for example, the manager may need to use appropriate interventions (such as wage increase, investment in new production technology or equipments, or further training) to boost the learning rate and hence steady-state productivity.

In this article we study the existence of steady-state skill level of workers in a batch production environment through the existence of fixed points. In the literature there are articles that consider similar problems. They either involve an experimental setting in which convergence to a unique value was observed (Sule, 1978, Globerson and Levin, 1987), or study a case of convergence to a unique value under specific functional forms of learning and forgetting (Axsater and Elmaphraby, 1981, Elmaghraby, 1990), or report a case of “alternating convergence” and investigate the average cost per period (Teyarachakul et al., 2008). The two cluster points case considered here, which is analogous to alternating convergence in the study of Teyarachakul et al. (2008), refers to a situation where the batch production time alternates between two distinct positive values in the steady-state.

Our analysis indicates that, under certain learning and forgetting environments, the batch production time may either converge to a fixed value (call it the desirable level) or alternate between two distinct points. In the latter case, the batch production times of workers tend to move away from the desirable level during the transition from one cycle to the next. Thus, in such a scenario, when skill level of workers is not at this desirable level, their productivity level will oscillate between these two distinct points, one below and one above the desired level. As in the previous case, this information can be put into use to help the managers make appropriate scheduling decisions. Furthermore, a manager who has this information, rather than being concerned of the workers’ unreliable productivity, will predict such an outcome and is likely to make necessary interventions. This could be in the form of training workers to the desired level at the beginning of each production cycle so that their skill level will not change (since it’s a fixed value). Furthermore, our convergence results are likely to provide deeper understanding long-term perspective of learning and forgetting effects on a number of manufacturing problems such as group scheduling (Yang & Chand, 2008), and flexibility acquisition policy (Kher, Malhotra, Philipoom, & Fry, 1999).

Although the work of Teyarachakul et al. (2008) is interesting in their discovery of a new type of convergence (referred to as alternating convergence there), their results are restricted to Globerson and Levin’s (1987) exponential forgetting function and Wright’s (1936) learning curve. There are large number of other well-performing predictors of learning and forgetting. A list of them can be found in Appendix A of Teyarachakul, Chand, and Ward (2011).

In this article we consider general classes of learning and forgetting functions; develop a mathematical model to analyze characteristics of learning and forgetting functions that lead to the convergence or clustering in batch production time. Our results extend the application and scope of the results in Teyarachakul et al. (2008) Furthermore, our analyses are quite deterministic; namely, given any pair of learning and forgetting functions (see Appendix A of Teyarachakul et al. (2011), or Example 1, Example 2, Example 3,Example 4, Example 5, Example 6, Example 7 in our Appendix A), in order to determine whether the convergence (to a single point) or two cluster point behavior is the case, all we need is to check if these functions satisfy the conditions (1), (2), and (3) in Section 2. Then Property 4.1, Property 4.2 will help in final identification of the type. Hence, the applicability of our results go beyond that of previous work of the same nature.

Besides providing a mathematical model describing and analyzing the behavior of workers’ skill levels, our results can also be utilized in predicting such patterns and develop appropriate actions/interventions. Namely, in a batch-production environment, where a manager has quite a good understanding of learning and forgetting trends of workers, by checking the parameters (1)–(3), he/she will know if workers’ skill level will converge or exhibit two-cluster point behavior. In the second case, the manager will likely to take some countermeasures to change the behavior to the convergence case, or will make use of the outcome to the benefit of the company in some way.

In Lapre, Mukherjee, and Van Wassenhove (2000), the authors studied, in part of their analysis, a learning curve that links different types of learning in quality improvement and reduction of waste rate. As a part of this study, they collected empirical data on the production volume spanning seven years beginning at 1984. The data shows alternating levels of production rates from one year to the next. Although they have a good idea on the learning curve, there is no information on the forgetting trend. Without any idea on the forgetting, one cannot firmly claim that the resulting production pattern is due to a similar skill level pattern. But, it is likely that they are linked. An inquisitive manager who wants to determine the cause(s) leading to such production patterns will seek some additional information. Our model and analysis provide an effective tool to identify some possible factors contributing to this end.

Any physical system that evolves in time, including the batch production environment under the investigation in this article, is an example of a dynamical system. Hence, it is natural to utilize the tools of dynamical systems in such a study. Fixed points of dynamical systems play a very important role in analyzing the long-term behavior of the states of a system; they are either attractive points of the system (to which all sequences in a certain neighborhood converge) or are repulsive points (from which all points, except the fixed point itself, move away). With these observations, this article will conduct investigations from the dynamical systems point of view; consequently, we provide a more rigorous approach to the steady-state results studied earlier (such as the ones in Teyarachakul et al. (2008)). In the meantime, we will also be able to study a wide variety of forgetting and learning functions simultaneously.

The organization of this paper is as follows. First, in Section 2 we formulate the model, as well as define functions and related variables. The model properties and convergence characteristics are presented in Sections 3 Properties of the functions, 4 Convergence results, respectively. In Section 5, we illustrate a scenario where case of two cluster points is observed, and how our model could be modified and be applied to other circumstances. Finally, we offer some final remarks and suggest future research opportunities. The appendix contains some selected examples for each of the convergence types studied.

Section snippets

Model formulation

In a batch production environment, the length of a production cycle is defined as the difference in the start times of two consecutive batches. Without loss of generality, we assume the production rate is greater than the demand rate; consequently, a production run is followed by an interruption between two successive batches to deplete the stored inventory. Workers learn during the production and forget during the break (a learn-forget cycle) within each production cycle. We will define skill

Properties of the functions G(x) and G2(x)

Our major interest is in the long-term behavior of the batch production time function, i.e. inlimkp(xk)=limkp(Gk-1(x)).

Naturally, this necessitates an in depth study of sequences of the form {Gk(x)}k=1. First, we note here that by the way G(x) is constructed, it must be a continuous function of x and so is each Gk(x).

The behavior of the sequence {Gk(x)} might follow numerous patterns. If the convergence of the sequence does not take place in a monotonic fashion, it turns out that the

Convergence results

In this section we will discuss the possible types of convergence and the convergence behavior of workers’ skill levels. The behavior of the convergence of the sequence {Gk(x)} is greatly influenced by the sign of G(x)x. For instance, if G(x)x0 for all x, then the sequence {Gk(x)} converges to x in a manner similar to the result obtained by Teyarachakul et al. (2008). Namely, if 0G(x)x, then x is an attractive fixed point (Davidson & Donsig, 2002); that is, when x is close enough to x

Discussion on possible scenarios and applications

In this section, we will discuss some real-world situations where: (a) one is likely to observe long-term batch production time alternate between two values as predicted by the results in Section 4, (b) one can utilize the information provided by our analysis (that batch production time may converge or may alternate between two values) in making managerial decisions, and (c) one can extend our results, with minor modifications to our model, to an environment with learning and forgetting in

Concluding remarks

This article considers convergence characteristics of workers’ skill levels under the presence of learning and forgetting in processing units. We study the steady-state behaviors under learning and forgetting for a more general class of functions; hence it generalizes Teyarachakul et al. (2008) work while extending the applicability of such results as described in Section 5. In doing so, we use the fixed-point approach, a well-known dynamical systems technique to study long-term behavior of

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