A learning curve for tasks with cognitive and motor elements

Computers & Industrial Engineering 64 (2013) 866–871

Contents lists available at SciVerse ScienceDirect

Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie

Technical Note

A learning curve for tasks with cognitive and motor elements Mohamad Y. Jaber a,⇑, Christoph H. Glock b a b

Department of Mechanical and Industrial Engineering, Ryerson University, 350 Victoria Street, Toronto, ON, Canada M5B 2K3 Industrial Management, Technische Universität Darmstadt, S1|03 85, Hochschulstr. 1, 64289 Darmstadt, Germany

a r t i c l e

i n f o

Article history: Received 27 December 2011 Received in revised form 20 July 2012 Accepted 2 December 2012 Available online 27 December 2012 Keywords: Learning curves Cognitive/motor elements Experimental data Fits EPQ model

a b s t r a c t This paper develops a new learning curve model that has cognitive and motor components. The developed model is fitted to experimental data of a repetitive manual assembly-and-disassembly task. The fits are compared to those of two other known models from the literature, which are the renowned power form learning curve and its aggregated version. The model developed in this paper performed the best. The fits of the models are evaluated using the mean squared error method. Furthermore, the developed learning curve model is investigated by incorporating it into the economic production quantity model, a topic which has been frequently studied by researchers. The results show that assuming an inappropriate learning curve may produce biased inventory policies by over- or underestimating production rates and consequently inventory levels. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Learning curves have been valuable management tools for decades. They predict and monitor the performance of individuals, groups of individuals and organizations. They have been widely used and applied in various sectors (manufacturing, healthcare, energy, military, information technologies, education, design, banking, and more). There are various forms of learning curves that are available in the literature (see for instance, Yelle, 1979; Hackett, 1983; Badiru, 1992; Badiru & Ijaduola, 2009; Jaber, 2006, chap. 32). All these models suggest that performance improves with repetition. Readers may refer to Jaber (2011) for theory, models and applications of learning curves. Of all the available models, the Wright (1936) learning curve remains to be the most popular (e.g., Yelle, 1979; Globerson, 1980; Badiru, 1992; Jaber, 2006, chap. 32). Its popularity is attributed to its simple mathematics and to its ability to fit a wide range of data fairly well (see Lieberman, 1987). Despite its popularity, the Wright learning curve, which is of a power form, has been criticized. For example, the results obtained from the Wright learning curve are not meaningful as the cumulative production approaches infinity. De Jong (1957) suggested introducing a plateauing factor to resolve this issue. Dar-El, Ayas, and Gilad (1995), based on evidence from the psychology and industrial engineering literature, suggested that the Wright learning curve is an aggregate learning curve that captures the cognitive and motor elements of a task or an experiment. To address this limitation, Dar-El et al. (1995) ⇑ Corresponding author. Fax: +1 416 979 5265. E-mail address: [email protected] (M.Y. Jaber). 0360-8352/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cie.2012.12.005

proposed the dual-phase learning curve model (DPLCM). Another limitation is that the Wright learning curve assumes that all units produced conform to quality. This assumption is unrealistic as many production processes are imperfect producing defective items that need to be reworked. Jaber and Guiffrida (2004) proposed a composite learning curve that is the sum of two learning curves; one describes the reduction in time for each additional unit produced while the other describes the reduction in time for each additional defective unit reworked. The composite learning curve model was found to have three behavioral patterns: Convex, plateau, and continuously decreasing. Only, the last behavior conformed with that of Wright (1936). Along the same line of research, this paper proposes a composite learning curve model that is similar to the dual-phase learning curve model. The developed model is fitted to experimental data taken from the study of Bailey (1989) and its fits are compared to those produced from the Wright learning curve and the learning curve of Dar-El et al. (1995). The data used in this study was collected in a laboratory experiment of a repetitive task that involved assembling and disassembling a mechanical apparatus performed by paid subjects. The assembly task was described by the author as being more complex than the disassembly task. In the course of Bailey’s study, workers were trained for 4–8 h in assembling and disassembling the apparatus. After a break of up to 4 months, the assembly-and-disassembly task was continued to analyze whether the subjects had forgotten some of the previously acquired skills. The assembly and disassembly process was overlooked and recorded by the researcher. For further information on the study, the reader is referred to the paper of Bailey (1989). In this study, the data of the learning sessions were used; i.e., the

M.Y. Jaber, C.H. Glock / Computers & Industrial Engineering 64 (2013) 866–871

data from the relearning (second) sessions were ignored as this study does not account for forgetting effects. This is beyond the scope of the paper and would be considered in a future work. The effects of learning on the economic order/production quantity (EOQ/EPQ) model (or the lot sizing problem) have been investigated frequently in the literature (e.g., Jaber & Bonney, 1999; Jaber & Bonney, 2011). Earlier studies assumed that learning in production may significantly reduce the total cost of an inventory system if a policy of producing smaller lots more frequently is adopted. Later studies investigated the combined or individual effects of learning in production, setups and quality on the lot sizing problem (Jaber & Bonney, 2003) or studied how learning influences the supplier selection decision (Glock, 2012), for example. The importance of learning in modern manufacturing and its effects on inventory policies have led some researchers to study its effects in a wider context; e.g., multistage production systems, supply chains and reverse logistics. Knowing how humans learn in production systems and how learning affects the performance of the production process is important for several reasons. For example, this enables production planners to assess how inventory develops over time, which is important to avoid bottlenecks in a production system. Further, knowing how much production capacity is available over time helps in drafting work plans and supporting the decision of whether peaks in demand need to be balanced by employing contractual (temporary) workers. Reader may also refer to Jaber (2011) for the importance of using the learning curve to investigate industrial engineering problems. We chose to investigate the applicability of the developed learning curve model in the context of the lot sizing problem. The next section provides a brief introduction of the Wright and Dar-El et al. learning curves. 2. The learning curves of Wright (1936) and Dar-El et al. (1995) The Wright (1936) learning curve is the earliest and the most popular model that depicts performance as a function of output. In addition, we note that the learning curve data used in this study is that of Bailey (1989), who found that the log-linear function fitted his data well, which is equivalent in fit results to the powerform curve of Wright (1936), and which is of the form log Tn = log T1  b log n. The Wright model advocates that performance reduces by a constant percentage each time the cumulative number of repetitions doubles. The Wright learning curve is of the form

T n ¼ T 1 nb ;

ð1Þ

where Tn is the time to produce the nth unit, T1 is the time to produce the first unit, n is the cumulative number of repetitions (units), and b is the learning exponent. The learning exponent is calculated as b = log (/)/log (2), where / is the learning rate and it is a percentage between 100% and 50% corresponding, respectively, to b = 0 and b = 1. For example, if T1 = 10 and the / = 0.8 (or 80%), then the time to produce the second, fourth, and eighth units are respectively T2 = 10  20.3219 = 8, T4 = 6.4, and T8 = 5.12. Dar-El et al. (1995) proposed an aggregate form of the Wright learning curve, the dual-phase learning curve model, where the resultant curve is the sum of two – cognitive and motor – learning curves. They assumed that both curves are of power forms like Wright’s. The dual-phase learning curve model (DPLCM) is of the form 

b bm T n ¼ ðT c1 þ T m ¼ T c1 nbc þ T m ; 1 Þn 1n

ð2Þ

where T c1 is the time to perform the first repetition under pure cognitive conditions, T m 1 is the time to perform the first repetition under pure motor conditions, bc is the learning exponent under pure cognitive conditions, and bm is the learning exponent under pure

867

motor conditions. The terms Tn and n were defined earlier. The learning exponent as observed after n repetitions was given as 

b ¼ bðnÞ ¼ bc 

logððR þ nbc bm Þ=ðR þ 1ÞÞ ; logðnÞ

ð3Þ

where R ¼ T c1 =T m 1 . In Dar-El et al. (1995), based on empirical data, the values of the learning exponents were taken as bc = 0.514 (/c = 70%) and bm = 0.152 (/m = 90%). However, they noted that further research could show that the values of the cognitive and motor exponents may differ from 0.514 and 0.152, respectively. 3. The proposed learning curve model It is assumed that a worker performing a task (e.g., assembling an item) will refer to procedure or steps prior to and during the execution of the task (e.g., looking up some information in a manual). That is, a portion of the time to perform each task will be to process information and acquire knowledge necessary to perform the task. For example, a worker operating on an assembly line where customized products are produced may have to refer to a manual or process description each time a product variant arrives at the workstation to look up how the production steps need to be performed. Similarly, in a job shop production process, workers may need to refer to manuals when changing from one job to the next to look up information on how to process the items in question. We refer to this process step as build-up of knowledge. After the worker has looked up the necessary information and read it in the manual, i.e. after knowledge has been built up, the production steps are performed. This process is referred to as the knowledge retrieval step. It is intuitively clear that learning may occur in both process steps: after the worker has produced product variants several times, it requires the worker less time to look up information about the production process, and the production process itself might as well be performed faster. The first effect is commonly described as cognitive learning, whereas the second effect is termed motor learning. The model proposed here is similar in form to that of Dar-El et al. (1995), and it is of the form

T n ¼ xT 1 nbc þ ð1  xÞT 1 nbm ¼ T 1 ½xðnbc  nbm Þ þ nbm ;

ð4Þ

where x is a percentage of splitting T1 into two components, cognitive and motor; i.e. T c1 ¼ xT 1 and T m 1 ¼ ð1  xÞT 1 . The cognitive component of Ti for repetition i e [1, n] reduces at a faster rate than the motor component. This is logical as a worker tends to recall a procedure or steps faster with every repetition, perhaps to the extent of instant recall, whereas the motor component of Ti, T m i , is much larger than the cognitive component, T ci , and possibly restricted to a lower bound. The reader may wonder if Eqs. (2) and (4) are the same, but they are not. Eq. (2) does not provide a mechanism of how T1 should be split between the motor and the cognitive component, while x in Eq. (4) is a model parameter that takes account of this fact. A second difference is that bm and bc in Eq. (2), or the DPLCM, are inputs and of fixed values, whereas bm and bc in JGLCM (Jaber-Glock learning curve model; Eq. (4)) are variables that are determined jointly with x to improve the model’s fit to data. 4. The fits In this section, the models presented in Section 2, WLC (Wright, 1936) and DPLCM (Dar-El et al., 1995), and the one presented in Section 3, which we will refer to as the JGLCM, are fitted to the experimental data of Bailey (1989). The experiment conducted by Bailey (1989) consisted of performing assembly tasks in a laboratory setting with students as surrogate workers in a manufacturing setting. Data for 102 learning curves were made available to us. To

868

M.Y. Jaber, C.H. Glock / Computers & Industrial Engineering 64 (2013) 866–871

fit the data to the JGLCM, the Mean Square Error (MSE) was minimized by optimizing for T1, x, bc, and bm. The MSE has been used and found by many to be a good measure of fits between estimated and actual learning data (e.g., Towill, 1977; Sikström & Jaber, 2012). The mathematical model is written as

Minimize MSE ¼

n 1X ðOi  T i Þ2 n i¼1

The data of Bailey (1989) was also fitted to the WLC using the least square method, since its simple mathematics allows that Eq. (1) can be written as log (Tn) = log (T1)  log(n). The data was fitted to DPLCM, similar to the model in Eqs. (5a)–(5e), by optimizing the following model

Minimize MSE ¼

ð5aÞ

n 1X ðOi  T i Þ2 n i¼1

ð6aÞ

Subject to:

Subject to:

bm P 0

ð5bÞ

bc  bm P 0

ð5cÞ

T1 P 0

ð5dÞ

0 6 x 6 1;

ð5eÞ

T c1 P 0

ð6bÞ

R P 0;

ð6cÞ

where Oi and Ti are the observed and estimated values of the learning curve for repetition i (from Eq. (3)), respectively, and where R ¼ T c1 =T m 1 . Note that Dar-El et al. (1995) set bc = 0.514 and bm = 0.152 corresponding to learning rates /c = 70% and /m = 90%, respectively. In the case of DPLCM, four fits per data set were performed for different sets of /c and /m values (Dar-El, 2000), which are: (1) 75%, 85%, (2) 72.5%, 87.5%, (3) 70%, 90%, and (4) 67.5%, 92.5%. The best of the four fits will be recorded; however, case 3 is recorded when it produces the same MSE value as the best case. The JGLCM performed better (for both the cases where bc 6 1 and where bc may exceed 1) than the WLC and the DPLCM. In this study, we dealt with 102 data sets. It is of course, for space constraints, not possible to tabulate the fits for all data sets, so we chose to only list the best five fits and the worst five fits of all models. The choice of these 10 fits would give the reader a reasonable idea about the best and worst performance of the three learning curve models. Table 1 summarizes the results of the best 5 and the worst 5 fits. The superiority of the JGLCM is due to its ability to predict the performance of the first few repetitions with much more accuracy than the other models, where the variation is higher between the actual data and the predicted performance. For example, the %deviation for all data in the prediction of T1 for the JGLCM was 1.12%, while it was 9.75% and 19.75% for the WLC (Wright, 1936) and DPLCM (Dar-El et al., 1995), respectively. Fig. 1 illus-

where Oi and Ti are the observed and estimated values of the learning curve for repetition i (from Eq. (4)), respectively. In this model, the cognitive learning exponent was assumed not to be restricted to an upper limit. Learning rates from many industries were reported to be in the range between 95% (corresponding to b = 0.074 in Eq. (1)) and 55% (corresponding to b = 0.862 in Eq. (1)); e.g., Cunningham (1980), Dutton and Thomas (1984), and Dar-El (2000). Salvendy (2001) noted that the initial time for a cognitive task may be 13–15 times the standard time, whereas the initial time for a manual task might be 2.5 times the standard time, corresponding to a maximum R value of 6 (15/2.5), which Dar-El et al. (1995, p. 265) noted in their paper. Values of bc larger than 1, although infrequent, were observed. For example, Camm (1985) noted that a colleague of his noticed learning curves in the 40% (b = 1.32)–50% (b = 1) percent range. Blancett (2002) also observed progress ratios in the range 1.1–2.0 for high-volume production lines. An empirical study (Hamade, Artail, & Jaber, 2005) conducted on CAD trainees observed learning rates faster than 50% (31%). The fits were repeated by adding a constraint, bc 6 1, to the problem described in (5a)–(5e).

Table 1 The best and worst five fits of the JGLCM, WLC, and DPLCM for the cases when bc P 0 and 0 6 bc 6 1. Data sets

0 < bc Best

n

JGLCM

WLC

T1

x

bc

bm

MSE

DPLCM

T1

b

MSE

T c1

Tm 1

R

bc

bm

b⁄

MSE

34 58 30 26 101

7 16 11 10 19

37.323 18.239 22.544 18.450 14.000

0.382 0.641 0.705 0.143 0.311

3.568 1.001 1.234 26.744 9.046

0.291 0.000 0.000 0.300 0.157

0.1136 0.1162 0.1294 0.1300 0.1642

32.653 15.817 19.020 17.462 11.224

0.510 0.319 0.430 0.357 0.221

5.0546 0.6396 1.6624 0.3301 0.6812

33.283 11.592 19.562 12.330 6.158

1.071 5.336 1.611 5.428 6.158

31.080 2.173 12.140 2.272 1.000

0.567 0.515 0.567 0.515 0.567

0.112 0.152 0.112 0.152 0.112

0.545 0.357 0.509 0.369 0.269

3.9877 0.5217 1.1108 0.3178 0.6101

15 8 19 54 10

3 7 5 11 8

73.503 47.583 67.516 22.784 83.941

0.000 0.000 0.161 0.000 0.717

0.979 2.838 0.822 25.211 1.251

0.660 0.300 0.822 0.448 0.000

56.2396 11.2177 9.3670 7.1661 6.6974

77.786 47.886 68.960 22.234 74.467

0.810 0.310 0.857 0.452 0.510

69.2802 11.2911 9.9845 7.2949 23.0363

71.113 24.808 59.195 19.967 76.852

0.000 22.775 0.000 2.828 1.969

VLN 1.089 VLN 7.062 39.030

0.567 0.567 0.567 0.515 0.567

0.112 0.112 0.112 0.152 0.112

0.567 0.300 0.567 0.448 0.549

61.4532 11.2177 40.3244 7.1661 17.9725

0 < bc < 1 Best 58 50 30 26 101

16 16 11 10 19

18.238 12.840 22.154 18.201 13.556

0.641 0.469 0.744 0.380 0.571

1.000 0.980 1.000 1.000 1.000

0.000 0.000 0.000 0.192 0.000

0.1162 0.1845 0.2136 0.2246 0.2257

15.817 11.387 19.020 17.462 11.224

0.319 0.183 0.430 0.357 0.221

0.6396 0.3966 1.6624 0.3301 0.6812

11.647 6.591 19.562 12.201 9.170

5.281 6.591 1.611 5.557 8.799

2.205 1.000 12.140 2.196 1.042

0.567 0.567 0.567 0.567 0.515

0.112 0.112 0.112 0.112 0.152

0.357 0.272 0.509 0.369 0.299

0.5217 0.6245 1.1108 0.3178 1.4250

3 8 7 5 9

73.503 39.233 47.583 67.516 42.186

0.000 0.851 0.000 0.161 0.949

0.979 1.000 0.946 0.822 1.000

0.660 0.000 0.300 0.822 0.000

56.2396 15.7282 11.2177 9.3670 8.4090

77.786 31.163 47.886 68.960 32.657

0.810 0.524 0.310 0.857 0.675

69.2802 26.7109 11.2911 9.9845 23.1162

71.113 33.582 24.808 59.195 33.330

0.000 0.815 22.775 0.000 0.000

VLN 41.181 1.089 VLN VLN

0.567 0.567 0.567 0.567 0.567

0.112 0.112 0.112 0.112 0.112

0.567 0.549 0.300 0.567 0.567

61.4532 23.1363 11.2177 40.3244 29.1005

Worst

Worst

15 35 8 19 40

VLN = Very Large Number. The bold values refer to minimum values when comparing the MSEs of JGLCM, WLC, and DPLCM.

869

M.Y. Jaber, C.H. Glock / Computers & Industrial Engineering 64 (2013) 866–871

Fig. 1. The difference in performance between the actual and predicted performance for the three learning curves.

trates, as an example, this behavior by plotting the difference between the actual performance (minutes) and the first set of the data (#1) for the WLC, DPLCM, and JGLCM.

0 6 bm < 1, and ours (III), where 0 6 cc < 1 and 0 6 bm < 1. The expressions of P’s for the three learning curve models are given, respectively, from Eqs. (1), (2), (4) as:

5. The economic production quantity (EPQ) model with learning effects

PI ðQ Þ ffi R Q 0

The EPQ model is a version of Harris’s (1913; reprinted 1990) celebrated model where the replenishment rate is finite (i.e., not instantaneous). The total cost per unit time function and the optimal production quantity (lot size) are given respectively as

  AD D Q D þ cp þ h 1 TCUðQ Þ ¼ Q P 2 P sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2AD ; Q ¼ hð1  D=PÞ 

  AD D Q D þ cp 1 þh ; Q Pi ðQ Þ 2 Pi ðQÞ

PII ðQ Þ ffi Q  ¼

Z 0

PIII ðQ Þ ffi Q ¼

ð9Þ

where Pi(Q) is the average unit production rate for learning curve i with Wright (I), where 0 6 b < 1, Dar-El (II), where 0 6 bc < 1 and

Z

(

ð8Þ

Q

¼

1b b Q ; T1

ð10Þ

1 bm ðT c1 nbc þ T m n Þdn 1

T c1 Tm 1 Q bc þ Q bm 1  bc 1  bm

ð7Þ

where A is the setup cost ($), P is the production rate measured in units per unit of time, D is the demand rate measured in units per unit of time (P > D), cp is the unit-time production cost, and h is the holding cost per unit per unit of time ($/unit/unit of time). The model also assumes an infinite planning horizon, no shortages, zero lead-time, all items produced conform to quality, the production and demand rates are constant over time, and all cost parameters do not vary over time as well. The lot sizing problem is the industrial engineering area were learning curves have most frequently been analyzed. The effects of learning on the lot sizing problem are reviewed in Jaber and Bonney (1999) and Jaber and Bonney (2011, chap. 14). Here, for illustrative purposes only, we use a simpler version than those models available in the literature and a modified version presented in Eq. (7). The reason for simplicity lies in the complexity in calculating the holding cost using Eqs. (2) and (4), as illustrated in the Appendix. The total cost per unit time function with learning effects can be approximated as

TCU i ðQ Þ ¼

Q T 1 nb dn

1

Q

T 1 ½xðnbc  nbm Þ þ nbm dn

0

"

T1 x

Q bc Q bm þ ð1  xÞ 1  bc 1  bm

ð11Þ

; 1

#)1

ð12Þ

Note that for model III, bc may take on the value of 1, wherefore Eq. (12) is rewritten as:

(

"

PIII ðQ Þ ffi Q T 1 x lnðQ Þ þ ð1  xÞ

Q 1bm 1  bm

#)1 :

ð13Þ

To illustrate, let us consider an inventory situation cp = $1000/day, A = 100, D = 12 units/day, and h is the holding = 0.1 $/unit/day. The inventory problem in Eq. (9) was optimized for the three models using the average production rate in Eqs. (10)–(13), using EXCEL Solver. The decision variable of the optimization problem is the lot size Q; the parameters of the learning curves are estimated with the help of the fits and are taken as given input parameters in the lot sizing problem. The lot size quantities where such that the values of Pi(Q) > D. The optimal lot size policies for four learning curve input parameters from Table 1 representing the best five and the worst five fits are considered. Of these fits, four were selected, which are the ‘‘best of best’’, #58, ‘‘worst of best’’, #101, ‘‘best of worst’’, #15, and ‘‘worst of worst’’, #40. The optimal production quantities and total costs for the three models for the four fits and the three models, JGLCM, WLC and DPLCM fits for the three models are tabulated below.

870

M.Y. Jaber, C.H. Glock / Computers & Industrial Engineering 64 (2013) 866–871

Table 2 The inventory policy for three learning curve models where 0 < bc 6 1. Data set#

WLC (Model I)

DPLCM (Model II)

JGLCM (Model III)

Q

TCU(Q)

Q

TCU(Q)

Q

TCU(Q)

58 101 15 40

483.51 417.77 698.67 477.18

32762.71 35808.24 27705.04 20764.70

417.80 492.11 861.37 544.70

36573.42 44197.97 41083.41 26100.85

226.90 205.80 761.15 324.63

55342.475 49348.142 33500.331 27846.075

The costs are annual; i.e., cost per day  365 assuming that 1 days is 10 h.

In Table 2, we only considered the case where 0 < bc < 1. Interested readers may refer to Jaber and Guiffrida (2007) for the effect of b > 1 on the lot size policy, where b is a combination of bc and bm. The results show that using models I and II biases the results, as III recommends producing in smaller lot sizes. Note that these results are for the very first cycle, with no prior experience. When several consecutive lots are produced and experience is gained, the results of the three models converge. To illustrate, for example T1 in Eqs. (10), (12), and (13) can be replaced by T1j = T1(1 + (j  1)Q)b (similarly, T k1j ¼ T j1 ð1 þ ðj  1ÞQ Þbk , where k = c, m in Eq. (11)), with (j  1)Q representing the cumulative experience with T1 = T11. As j increases, Eqs. (10)–(13) reach a very large value (instantaneous replenishment), where Eq. (9) for all models converges to the EOQ formula, TCUðQ Þ ¼ AD þ h Q2 (see Jaber and Bonney Q (1999), Jaber and Bonney (2011, chap. 14)).

model to use. Further, comparing the models studied in this paper in a more comprehensive study, and with the help of data gained from learning processes in different work environments would be interesting. The developed model is promising and it would be interesting to modify it to account for forgetting. That is, it could be tested against the learn-forget curve model (Jaber & Bonney, 1996) and the dual-phase learning-forgetting model (Jaber & Kher, 2002), which are direct extensions of the learning curves of Wright (1936) and Dar-El et al. (1995). Acknowledgements The first author sincerely thanks Prof. Charles D. Bailey, the Arthur Andersen Chair of Excellence and Professor of Accountancy at The University of Memphis, for making his data available to us. Without it, this study would not have been possible. The authors thank the Natural Sciences and Engineering Council of Canada (NSERC) for their financial support. The second author thanks the Department of Mechanical and Industrial Engineering at Ryerson University for the in-kind support provided during his stay as a visiting researcher. He further thanks the German Academic Exchange Service (DAAD) for supporting his work by a fellowship within the Postdoc-Program.

Appendix A 6. Conclusions In this paper, a new learning curve model (JGLCM) was developed. It is an aggregated curve of cognitive and motor components. The developed model was tested against empirical data from Bailey (1989). The fits showed that the JGLCM performed the best when these fits were compared to those of two popular learning curve models from the literature; namely, the WLC (Wright, 1936) and DPLCM (Dar-El et al., 1995). The three models were also investigated in an inventory setting. The results suggest that using either the Wright or Dar al et al. learning curves may bias the values of the lot size quantity and the total cost. On a separate note, EXCEL SOLVER was used to determine the fits by solving a mathematical programming model with the mean-square error as the objective function. Although Excel Solver is widely used by researchers, some may argue that it is not that effective when dealing with non-linear optimization problems. For the sake of argument, if Excel Solver produces an error in optimization, the error would be consistent for all three learning curve models, which are JGLCM, WLC and DPLCM, of which the JGLCM fits were better than the remaining others. This does not qualify the JGLCM from being a promising model. A final note on this issue, random samples from Bailey’s data were selected and fitted using Mathematica 8.0 by Wolframs Research and the results were shown to be consistent with those produced by Excel Solver. The study presented in this paper has limitations. When interpreting the results, it has to be considered that learning processes depend on a variety of different factors, such as the characteristics of the individual, the environment the individual works in, or the characteristics of the tasks that are performed, just to name a few. Thus, we may assume that the precision with which a learning curve describes a learning process varies with the characteristics of the learning process as well. The fact that the JGLCM outperformed the WLC and the DPLCM in this paper does therefore not necessarily mean that it will always outperform these models and that it should therefore be preferred to them in principle. Instead, we recommend that researchers who wish to use one of these models evaluate the situation under study in detail and then decide which

The holding cost per cycle for the EOQ/EPQ model is determined by integrating the inventory level equation, i.e. I(t) = Q(t)  Dt, over the proper limits, where



0 6 t 6 t p ¼ Q =P

ðP  DÞt;

IðtÞ ¼

ðP  DÞt p  Dt; 0 6 t 6 t I ¼ ðP  DÞQ =PD

ðA:1Þ

and tp is the time to produce Q units and accumulate a maximum inventory of (1  D/P)Q in a cycle, and tI is the time to deplete the maximum inventory (cycle time = Q/D = tp + tI). The holding cost is then calculated as

h

Z

Q =P

ðP  DÞtdt þ h

0

Z

ð1D=PÞQ=D

ðð1  D=PÞQ  dtÞ dt

0

h PD 2 ðP  DÞð1  D=PÞ h ð1  D=PÞ2 2 Q  Q þ h Q 2 P2 Pd 2 D   2 h D Q 1 ¼ ; 2 P D ¼

ðA:2Þ

  with the holding cost per unit of time as 2h 1  DP Q . The total time to produce Q units by using Eq. (4) is calculated as

tðQ Þ ¼

Z

Q

T 1 ðxnbc þ ð1  xÞnbm Þdn

0

¼

xT 1 ð1  xÞT 1 1bm Q 1bc þ Q 1  bc 1  bm

ðA:3Þ

One would notice that writing Q(t), and subsequently I(t) = Q(t)  Dt, is not possible. This also applies for Eq. (2). Therefore, an approximation of the holding cost expression is necessary (see, Jaber and Bonney, 2011, chap. 14). Using an approximate model would produce reasonable results as illustrated in the numerical example blow. Salameh et al. (1993), who investigated the effects of learning on the lot sizing problem, proposed the following cost function

" # AD T 1j DQ b Q T 1j DQ 1b ; þ cp  TCUðQ Þ ¼ þh Q 2 ð1  bÞð2  bÞ 1b

ðA:4Þ

M.Y. Jaber, C.H. Glock / Computers & Industrial Engineering 64 (2013) 866–871

where T1j = T1(1 + (j  1)Q)b with (j  1)Q is the experience count. For example, Salameh et al. (1993) assumed learning to follow the WLC. In this paper, for the same learning curve (i = I), we have (from Eqs. (9) and (10))



AD D Q D þ cp 1 þh Q PI ðQ Þ 2 P I ðQ Þ " # b AD T 1j DQ Q T 1j DQ b þ cp 1 þh ¼ Q 2 1b 1b

TCU I ðQ Þ ¼

ðA:5Þ

Let us consider the data T1 = 1/16, D = 12, A = 200, cp = 1000, h = 0.1, and b = 0.1. Eqs. (A.4) and (A.5) were optimized using Excel Solver. The optimal solutions for Eqs. (A.4) and (A.5) occur at: (1) TCU(Q) = 446 with Q = 1370, and (2) TCUI(Q) = 447 with Q = 1335, respectively, with an error that is less than |0.25|% in the cost. The problem was investigated for b = 0.5, the results are: (1) TCU(Q) = 92.4 with Q = 675, and (2) TCUI(Q) = 93.1 with Q = 670, respectively, with an error that is less than |0.8|% in the cost. Assuming a faster initial production rate (1/T1), the results for b = 0.5 and T1 = 1/160 are: (1) TCU(Q) = 31.35 with Q = 270, and (2) TCUI(Q) = 31.39 with Q = 270, respectively, with an error that is less than |0.15|% in the cost. Note that a higher production rate is similar to an increase in experience, i.e., an increase in j. From the above, one may conclude that the approximation presented for the inventory cost function, Eq. (9), is reasonable and will satisfactorily do its indented illustrative purpose in this paper. References Badiru, A. B. (1992). Computational survey of univariate and bivariate learning curve models. IEEE Transactions on Engineering Management, 39(2), 176–188. Badiru, A. B., & Ijaduola, A. O. (2009). Half-life theory of learning curves for system performance analysis. IEEE Systems Journal, 3(2), 154–165. Bailey, C. D. (1989). Forgetting and the learning curve: A laboratory study. Management Science, 35(3), 346–352. Blancett, R. S. (2002). Learning from productivity learning curves. Research Technology Management, 45(3), 54–58. Camm, J. D. (1985). A note on learning curve parameters. Decision Sciences, 16(3), 325–327. Cunningham, J. A. (1980). Using the learning curve as a management tool. IEEE Spectrum, 17(6), 43–48. Dar-El, E. M. (2000). Human learning: from learning curves to learning organizations. Boston/Dordrecht/London: Kluwer Academic Publishers. Dar-El, E. M., Ayas, K., & Gilad, I. (1995). A dual-phase model for the individual learning process in industrial tasks. IIE Transactions, 27(3), 265–271. De Jong, J. R. (1957). The effect of increased skills on cycle time and its consequences for time standards. Ergonomics, 1(1), 51–60.

871

Dutton, J. M., & Thomas, A. (1984). Treating progress functions as a managerial opportunity. Academy of Management Review, 9(2), 235–247. Globerson, S. (1980). The influence of job related variables on the predictability power of three learning curve models. AIIE Transactions, 12(1), 64–69. Glock, C. H. (2012). Single sourcing versus dual sourcing under conditions of learning. Computers & Industrial Engineering, 62(1), 318–328. Hackett, E. A. (1983). Application of a set of learning curve models to repetitive tasks. The Radio and Electronic Engineer, 53(1), 25–32. Hamade, R. F., Artail, H. A., & Jaber, M. Y. (2005). Learning theory as applied to mechanical CAD training of novices. International Journal of Human–Computer Interaction, 19(3), 305–322. Harris, F. W. (1990). How many parts to make at once? Operations Research, 38(6), 947–950 (Reprinted from Factory: The Magazine of Management 10(2), 1913, pp.135–36). Jaber, M. Y. (Ed.). (2011). Learning curves: Theory, models, and applications. FL, Baco Raton: CRC Press (Taylor and Francis Group). Jaber, M. Y. (2006). Learning and forgetting models and their applications. In A. B. Badiru (Ed.), Handbook of industrial & systems engineering (pp. 1–27). FL, Baco Raton: CRC Press – Taylor & Francis Group. Jaber, M. Y., & Bonney, M. (1996). Production breaks and the learning curve: The forgetting phenomena. Applied Mathematical Modelling, 20(2), 162–169. Jaber, M. Y., & Bonney, M. (1999). The economic manufacture/order quantity (EMQ/ EOQ) and the learning curve: Past, present, and future. International Journal of Production Economics, 59(1–3), 93–102. Jaber, M. Y., & Bonney, M. (2003). Lot sizing with learning and forgetting in set-ups and in product quality. International Journal of Production Economics, 83(1), 95–111. Jaber, M. Y., & Guiffrida, A. L. (2004). Learning curves for processes generating defects requiring reworks. European Journal of Operational Research, 159(3), 663–672. Jaber, M. Y., & Bonney, M. (2011). The lot sizing problem and the learning curve: A review. In M. Y. Jaber (Ed.), Learning curves: Theory and models, and applications (pp. 267–294). FL, Baco Raton: CRC Press – Taylor and Francis Group. Jaber, M. Y., & Guiffrida, A. L. (2007). Observations on the economic order (manufacture) quantity model with learning and forgetting. International Transactions in Operational Research, 14(2), 91–104. Jaber, M. Y., & Kher, H. V. (2002). The two-phased learning-forgetting model. International Journal of Production Economics, 76(3), 229–242. Lieberman, M. B. (1987). The learning curve, diffusion, and competitive strategy. Strategic Management Journal, 8(5), 441–452. Salameh, M. K., Abdul-Malak, M. A., & Jaber, M. Y. (1993). Mathematical modelling of the effect of human learning in the finite production inventory model. Applied Mathematical Modelling, 17(11), 613–615. Salvendy, G. (2001). Handbook of industrial engineering: Technology and operations management.. New York: Institute of Industrial Engineering, Wiley. Sikström, S., & Jaber, M. Y. (2012). The Depletion–Power–Integration–Latency (DPIL) model of spaced and massed repetition. Computers & Industrial Engineering, 63(1), 323–337. Towill, D. R. (1977). Exponential smoothing of learning curve data. International Journal of Production Research, 15(1), 1–15. Wright, T. (1936). Factors affecting the cost of airplanes. Journal of Aeronautical Science, 3(2), 122–128. Yelle, L. E. (1979). The learning curve: Historical review and comprehensive survey. Decision Sciences, 10(2), 302–328.