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Int. J. Production Economics 108 (2007) 359–367

www.elsevier.com/locate/ijpe

Economic manufacture quantity (EMQ) model with lot-sizedependent learning and forgetting rates

Mohamad Y. Jabera,�, Maurice Bonneyb

aDepartment of Mechanical and Industrial Engineering, Ryerson University, Toronto, ON, M5B 2K3 CanadabNottingham University Business School, Jubilee Campus, Nottingham NG8 1BB, UK

Received 11 August 2004; accepted 11 October 2005

Available online 25 January 2007

Abstract

Optimal lot-size formulae typically assume that unit manufacturing costs are constant. This is not true while learning is

in progress. Many researchers have studied the effect of learning on the lot-size problem. These works commonly assume

an invariant learning slope throughout the production-planning horizon. When learning rates (LRs) are dependent on the

number of units produced in a production cycle, then the assumption of invariant LRs might produce erroneous lot-size

policies. This paper investigates the effect of lot-size dependent learning and forgetting rates on the lot-size problem by

incorporating the dual-phase learning–forgetting model (DPLFM) [Jaber, M.Y., Kher, H.V., 2002. The dual-phase

learning-forgetting model. International Journal of Production Economics 76(3), 229–242] into the economic manufacture

quantity model. Numerical examples are provided to illustrate the solution procedure. These compare different approaches

to lot sizing with learning and forgetting effects.

r 2007 Elsevier B.V. All rights reserved.

Keywords: EMQ; Learning; Forgetting; Variable learning and forgetting rates

1. Introduction

How to determine the economic manufacturequantity (EMQ) has been a challenge to researchersand practitioners for many decades. The standardEMQ model based on the lot-size formula (Harris,1915) has been subjected to many extensions toreflect real-world situations better, which includeinvestigating the EMQ when there is learning andforgetting in production. The earliest such work isbelieved to be that of Keachie and Fontana (1966).

front matter r 2007 Elsevier B.V. All rights reserved

e.2006.12.020

ng author. Tel.: +416 979 5000x7623;

265.

ss: mjaber@ryerson.ca (M.Y. Jaber).

This triggered several researchers to investigate thelot-sizing problem with learning, and learning andforgetting, a survey of which appeared in Jaber andBonney (1999).

One limitation of the EMQ model is the assump-tion that the unit manufacturing cost is constant.When the manufacturing process undergoes learn-ing in production, the unit cost (time per unit)decreases with cumulative output. This relationshipis in conformance with that advocated by thelearning curve of Wright (1936) described morefully in Section 2. Although some researchers (e.g.,Zangwill and Kantor, 1998; Terwiesch and Bohn,2001) have criticised modelling time per unit as afunction of cumulative production, they have not

.

ARTICLE IN PRESSM.Y. Jaber, M. Bonney / Int. J. Production Economics 108 (2007) 359–367360

proposed that cumulative production should beexcluded. Indeed, most investigations of the lot-sizing problem with learning and forgetting haveassumed that learning in production conforms tothat of Wright (1936) or the similar form of DeJong(1957). Though learning curves have been used andinvestigated for almost seven decades now, someresearchers believe that they may be used morewidely in the future due to the demand forsophisticated high-technology systems (Steven,1999) and because the increased rate of productand process change means that many products areproduced in the early stages of learning. If so,estimating the effects of learning is becoming moreimportant.

Jaber and Bonney (1999) categorised publicationson lot sizing and learning, and learning andforgetting, into two groups. The first group assumedinstantaneous replenishment of inventory whereasthe second group assumed a finite delivery rate.Most of the work surveyed assumed that learningconforms to that of Wright (1936), although Fiskand Ballou (1982) and Jaber and Bonney (1996a)investigated the EMQ model when learning con-forms to that of DeJong (1957). A commonassumption among the work surveyed in Jaberand Bonney (1999) is that there is an invariantlearning slope throughout the production-planninghorizon. When learning rates (LRs) are dependenton the number of units produced in a productioncycle (Dar-El et al., 1995a), then the assumption ofinvariant LRs could produce biased lot-size policies.Therefore, in this paper, we incorporate the dual-phase learning–forgetting model (DPLFM) devel-oped by Jaber and Kher (2002) into the EMQmodel. The DPLFM assumes that the learningand forgetting rates are dependent on the lot-sizequantity.

This paper and those surveyed in Jaber andBonney (1999) assume that a single product isproduced on a single machine with no set-up time.A more realistic and a possible extension to thework presented here is to consider productionand set-up times for a capacity constrained facilitywith multiple products such as discussed bySegerstedt (1999), but in the presence of learningand forgetting.

The remainder of this paper is organised asfollows. Section 2 provides the rationale for adopt-ing the DPLFM. Section 3 briefly describes themathematics necessary for the DPLFM. Section 4describes the mathematical modelling that incorpo-

rates DPLFM into the EMQ model. Section 5provides numerical examples, compares the resultswith those derived using an EMQ model thatassumes an invariant LR (Salameh et al., 1993),and discusses the results. Section 6 summarises theconclusions.

2. The learning and forgetting models

This section provides a brief introduction to thelearning and forgetting models adopted in thispaper. More extensive discussion of past and recentdevelopments in modelling the learning–forgettingprocess appears in Jaber (2006).

2.1. The learning model

The earliest learning curve representation is ageometric progression that expresses the decreasingcost (or time) required to accomplish any repetitiveoperation. The theory states that as the totalquantity of units produced doubles, the cost perunit declines by some constant percentage. Wright’s(1936) learning curve is a power function formula-tion that is represented by

Tx ¼ T1x�b, (1)

where Tx is the time to produce the xth unit, T1 thetime to produce the first unit, x the productioncount, and b the learning curve exponent. Inpractice, the b parameter value is often replacedby another index number that has had moreintuitive appeal. This index, called the ‘‘learningrate’’ (LR), is the ratio of the unit production timeswhen the output is doubled, where LR ¼ T2x/Tx ¼

T1(2x)�b/T1x�b¼ 2�b and b ¼ �log(LR)/log(2).

For example, if a production process has an 80%LR, b ¼ �log(LR)/log(2) and so b ¼ �log(0.8)/log(2) ¼ 0.3219. If the time to produce the first unitis 100min (T1 ¼ 100), then the time to produce thesecond unit is T2 ¼ 100(2)�0.3219 ¼ 80min, thefourth unit T4 ¼ 100(4)�0.3219 ¼ 64min, the 100thunit T100 ¼ 100(100)�0.3219 ¼ 22.71min, etc.

2.2. The forgetting model

The dual-phase learning model (DPLM) pro-posed by Dar-El et al. (1995a) suggests that learningan industrial task, like all daily tasks, consists of twoparallel phases: cognitive and motor. The behaviourof the DPLM is illustrated in Fig. 1. This model wasdeveloped when their carefully generated research

ARTICLE IN PRESS

y (1)

yc(1)

ym(1) Motor

Cognitive

Wright’s learning curve

Number of repetitions

Time per repetition

Fig. 1. Wright’s learning curve as a combined curve of two

learning curves: cognitive and motor (Dar-El et al., 1995a, b).

M.Y. Jaber, M. Bonney / Int. J. Production Economics 108 (2007) 359–367 361

data were found to match poorly all the knownmathematically based learning curve models. Dar-El et al. (1995a) collected data from an experimentalapparatus of assembling electric matrix board andelectronic components. Dar-El et al.’s experimentinvolved two tasks: one complex (more use ofcognitive skills and less use of motor skills) and onesimple (less use of cognitive skills and more use ofmotor skills). They concluded that the learningslope did not have a fixed value; it gradually reducesas experience is gained. In a subsequent paper, Dar-El et al. (1995b) investigated the DPLM forforgetting. They experimentally investigated therelationship between the learning exponent withforgetting, bf and without forgetting, b0. Theirresults indicated that bf follows a power decayform, bf ¼ b0(1+q)�0.152 where q is the interruptionperiod in days. Dar-El et al. (1995b, p. 279) furtheradded that there were insufficient data to fullysupport the relationship between bf and b0.

The tendency in industry is to automate mechan-ical tasks as much as possible and to utilise peoplefor control and decision-making (Dar-El, 2000,p. 87). Similarly, Nembhard and Uzumeri (2000)observed that many workplace tasks are beingautomated and, as a result, the remaining tasksare more cognitive and complex and difficult toautomate. Thus, a learning–forgetting model thatcaptures the motor and cognitive elements ofindustrial tasks is desirable, especially, in manufac-turing environments that emphasise workforceflexibility where production is repetitive and usuallyin small batches (e.g., Wisner and Siferd, 1995).

Jaber and Kher (2002) developed the dual-phaselearning–forgetting model (DPLFM) by combiningthe DPLM (Dar-El et al., 1995a) with the learn–forget curve model (LFCM; Jaber and Bonney,1996b). The LFCM captures forgetting effects in a

manner that is consistent with empirical observa-tions concerning forgetting (Jaber and Bonney,1997; Jaber et al., 2003) and was found to fitempirical data well (Jaber and Sikstrom, 2004). TheDPLFM retains the desirable characteristics of boththe DLFM and LFCM discussed above, and thuscan help managers develop more reliable estimatesof task times than the ones provided by either modelalone. The DPLFM is adopted in this paper and itsmathematics is presented in the next section.

3. The mathematics of the dual-phase

learning–forgetting model (DPLFM)

Jaber and Kher (2002) combined the DPLM(Dar-El et al., 1995a) with the LFCM (Jaber andBonney, 1996b) to develop the DPLFM. All threemodels: DPLM, LFCM, and DPLFM assume thatthe LR conforms to that of Wright (1936).

Jaber and Kher (2002) defined Dc and Dm as thetime for total forgetting for cognitive and motorcomponents of learning, respectively. Similarly, theydefine yc

i ð1Þ; ymi ð1Þ and fi

c, fim as the intercepts and

slopes for cognitive and motor forgetting curves,respectively. Jaber and Kher (2002) proposed aforgetting function of the form

yið1Þnf i

i ¼ yci ð1Þn

f ci

i þ ymi ð1Þn

f mi

i , (2)

where yið1Þ ¼ yci ð1Þ þ ym

i ð1Þ, 0pficp1, 0pfi

mp1,0pfip1, and ni is the number of units producedin cycle i. The forgetting slopes for the cognitive andmotor forgetting curves are, respectively,

f ci ¼

bcð1� bcÞ logðuci þ niÞ

logðCci þ 1Þ

and (3a)

f mi ¼

bmð1� bmÞ logðumi þ niÞ

logðCmi þ 1Þ

, (3b)

where the terms uic and ui

m reflect the cognitive andmotor experience retained from the previous i�1cycles at the start of cycle i, respectively. The terms Ci

c

and Cim represent the ratio of total time to forgetting

and the time needed to acquire (uic+ni) and (ui

m+ni)units of cognitive and motor experience, respectively.The expressions for Ci

c and Cim are given by

Cci ¼ Dc

ycð1Þðuci þ niÞ

1�bc

ð1� bcÞ

" #�1, (4a)

Cmi ¼ Dm

ymð1Þðumi þ niÞ

1�bm

ð1� bmÞ

" #�1. (4b)

ARTICLE IN PRESSM.Y. Jaber, M. Bonney / Int. J. Production Economics 108 (2007) 359–367362

After an interruption of length d in cycle i�1, theperformance of the cognitive and motor learningcurves will be given by ~yc

i ð1Þ and ~ymi ð1Þ as

~yci ð1Þ ¼ yð1Þðuc

i þ 1Þ�bc , (5a)

~ymi ð1Þ ¼ yð1Þðum

i þ 1Þ�bm. (5b)

The number of units remembered (i.e., experienceretained) ui is given by

uðiÞ ¼~yc

i ð1Þ þ ~ymi ð1Þ

yð1Þ

� ��1= ~biðniÞ

� 1, (6)

where ~yið1Þ ¼ ~ymi ð1Þ þ ~yc

i ð1Þ is the time to produce thefirst unit after an interruption. The learning exponent~biðnÞ is given as

~biðniÞ ¼ bc �logððRþ ðui þ niÞ

bc�bm Þ=ðRþ 1ÞÞ

logðui þ niÞ(7)

where, and as defined by Dar-El et al. (1995a),R ¼ yc(1)/ym(1), is the ratio of the time for the firstrepetition under purely cognitive conditions, yc(1), tothe time for the first repetition under purely motorconditions, ym(1), and R ¼ yc(1)/ym(1) is constant fora given task (Dar-El, 2000, p. 47). In the next section,the DPLFM is incorporated into the EMQ model.

4. The mathematical model

Define K as the set-up cost per period, h as theholding cost per unit per unit time, L as the labourcost per unit time, and r as the demand rate. Thetotal cost of producing and depleting qi units inperiod i, C(qi), is the sum of the production cost,P(qi), and holding cost, H(qi). P(qi) is the sum of theset-up cost and the total production cost to producea lot of size qi, where P(qi) ¼ K+Lt(qi), and tðqiÞ ¼Pqi

n¼1yið1Þn�biðqiÞis the time to produce a lot of size

qi. Similarly, as in Salameh et al. (1993), theproduction and holding costs are determined,respectively, as

PðqiÞ ¼ K þ LXqi

n¼1

~yið1Þn�biðqiÞ ffi¼ K þ

L ~yið1Þqð1�biðqiÞÞ

i

1� biðqiÞ.

(8)

The approximation of the expression LPqi

n¼1yið1Þn�biðqiÞ in (8) was agreed by many researchers, (e.g.,Keachie and Fontana, 1966), to be good. t(qi) isdefined as the time to build a maximum inventory ofZi units in production cycle i, where Ii(t) ¼ qi(t)�rt

is the inventory level over the interval 0ptpt(qi)and Ii(t ¼ t(qi)) ¼ Zi. Once an inventory level of size

Zi is attained, production ceases, and the Zi unitsare depleted at a constant demand rate of r units perunit of time. At the end of the period t(Zi); wheret(Zi) ¼ Zi/r and Ii(t) ¼ �rt+rTi is the inventorylevel over the interval t(qi)ptpTi and Ti is the cycletime for cycle i; the inventory level is zero; i.e.,Ii(t ¼ Ti) ¼ �rt+rTi ¼ 0 and Ti ¼ t(qi)+t(Zi) ¼qi/r. By integrating Ii(t) over the proper limits, theholding cost could be written as [see Eqs. (4)–(10) inSalameh et al. (1993, p. 614), for detailed mathe-matics]

HðqiÞ ¼hq2

i

2r�

h ~yið1Þqð1�biðqiÞÞ

i

ð1� biðqiÞÞð2� biðqiÞÞ. (9)

The total cost per period is the sum of (8) and (9),i.e. C(qi) ¼ P(qi)+H(qi). This is given by

CðqiÞ ¼ K þL ~yið1Þq

½1�biðqiÞ�

i

1� biðqiÞþ

hq2i

2r

�h ~yið1Þq

½2�biðqiÞ�

i

ð1� biðqiÞÞð2� biðqiÞÞ, ð10Þ

where ro1/y(1). Then the cost per unit of time,c(qi) ¼ C(qi)/Ti, and referred to in this paper by theJaber and Bonney model (JBM) could be written as

cðqiÞ ¼ Kr=qi þL ~yið1Þ

1� biðqiÞrq�biðqiÞ

i þ hqi

2

�h ~yið1Þrq

½1�biðqiÞ�

i

ð1� biðqiÞÞð2� biðqiÞÞ. ð11Þ

To show that (11) holds a unique minimum at qi*X1,two observations are made. First, Eq. (11) ap-proaches Kr+Lyi(1)r/(1�bi(1)) as qi approaches 1,where bi(1) ¼ bc�(bc�bm)/(Ri+1) from Eq. (7).Eq. (11) approaches infinity as qi approaches infinity,where by applying l’Hopital’s rule to Eq. (7),limqi!1

~bðqiÞ ¼ bc and (11) is positive for bothobservations. Second, the limit of the first derivativeof (11) as qi approaches 1 from the right-hand side is

limqi!1þqcðqiÞ=qqi ¼ �Kr� L ~yið1Þbið1Þ=ð1� bið1ÞÞ

� hbið1Þ=2ð2� bið1ÞÞo0; qbðqiÞ=qqi

��qi¼1¼ 0.

This indicates that (10) has a global minimum atqi*, where c(qi*+1)4c(qi*) and c(qi*�1)4c(qi*).

5. Numerical examples

This section illustrates by means of numericalexamples the behaviour of the model described inSection 4 for various cognitive and motor LRs.Dutton and Thomas (1984) showed the distribution

ARTICLE IN PRESSM.Y. Jaber, M. Bonney / Int. J. Production Economics 108 (2007) 359–367 363

of LRs for 108 firms, with a mean of 80.11%(b ¼ 0.311). Of the 108 firms, the LRs for 93 firmswere in the range of 71–89% (0.4944b40.168).Dar-El (2000) summarised the LRs for specific typesof work from observations by several researchers,which ranged from 68% for a truck body assemblyto 95% for raw materials usage (see Table 4.1 inDar-El, 2000, pp. 58–59). Dar-El et al. (1995b,p. 273) classified LRs into four categories, with fbeing the LR as defined by Wright (1936). The fourcategories are high cognitive (70ofp75%), more

cognitive than motor (75ofp80%), more motor

than cognitive (80ofp85%), and high motor

(85ofp90%). Based on the above, and followingthe classification of Dar-El et al. (1995b, p. 273) forcognitive (where 70% is high and 80% is low) andmotor (where 90% is high and 80% is low) LRs, thefollowing situations of cognitive and motor LRs areadopted in this study, which are: (1) high cognitiveand high motor LRs (fc ¼ 70%, fm ¼ 90%), (2)medium cognitive and medium motor LRs(fc ¼ 75%, fm ¼ 85%), and (3) low cognitive andlow motor LRs (fc ¼ 80%, fm ¼ 80%). Thesecombinations would produce a combined LR ~fiðnÞ

(associated with ~biðnÞ in Eq. (6), where fiðnÞ ¼

100� 2�~biðnÞ) in the range of 90–70% as described in

Dar-El (2000, pp. 58–59). Note that values offc480% and fmo80% are outside the range ofDar-El’s investigation and are not considered.

Consider an inventory situation where the follow-ing conditions apply: the set-up cost K ¼ 200, theholding cost h ¼ 0.05/unit/day, the demand rater ¼ 12 units/day, the production cost L ¼ 100/day,and the time to process the first unit y1 ¼ 1/16 days/unit. As stated earlier, when LRs are dependent onthe number of units produced in a production cycle,then the assumption of invariant LRs mightproduce erroneous lot-size policies. To comparethe model developed in Section 4, which assumes avariable LR, with the model of Salameh et al.(1993), which assumes a fixed LR, it is necessary tofind the equivalent learning slope for the latermodel. To do this, we use the following relationship

b ¼ �log

yc1ð2Þ�bcþym

1ð2Þ�bm

y1

h ilogð2Þ

, (12)

where yðnÞ ¼ yc1n�bc þ ym

1 n�bm is the form of theDPLM (Dar-El et al., 1995a), and y(n) ¼ y1n

�b isthe form of the Wright’s learning curve (Wright,1936) when n ¼ 2. For example, for combination 1where fc ¼ 70% (bc ¼ 0.515) and fm ¼ 90%

(bm ¼ 0.152) and R ¼ yc(1)/ym(1) ¼ 1, i.e. yc(1) ¼ym(1) ¼ 1/32, the learning slope to be adopted inSalameh et al.’s model (SAJM) is computed from(11) as

b ¼ �log ð1=32Þð2Þ

�0:515þð1=32Þð2Þ�0:152

ð1=16Þ

h ilogð2Þ

¼ 0:322,

which corresponds to an 80% LR. In Salameh et al.(1993) the LR was assumed to be a fixed value. Thecost model of Salameh et al. (1993) was modified byJaber and Bonney (1996b) to account for forgettingto give

cðqiÞ ¼Kr

qi

þL ~yið1Þ

1� brq�b

i þ hqi

2�

h ~yið1Þrq1�bi

ð1� bÞð2� bÞ,

(13)

where ~yið1Þ ¼ yð1Þð1þ uiÞ�b, and ui is the number of

units remembered (i.e., experience retained) from

the previous i�1 cycles; such that 0ouioPi�1

n¼1qn.

Note that when there is no forgetting, ui ¼Pi�1

n¼1qn,

and (12) reduces to the cost model of Salameh et al.(1993). The cost expression in (13) will be referred toin this paper by the SAJM.

For the above inventory model parameters,learning combinations, R ¼ 1/3, 1, and 3, and thecase of no forgetting (Dc-N and Dm-N), thecost functions described in (11) (JBM) and (13)(SAJM) are optimised for 10 consecutive cycles.Table 1 summarises the results for the case whenR ¼ 1, fc ¼ 70% (bc ¼ 0.515), and fm ¼ 90%(bm ¼ 0.152).

The results in Table 1 show that not accountingfor the cognitive and motor elements of a job mayproduce erroneous results, that is using SAJM onthe average, under-estimates the cost by about 20%((18.30�22.83)/22.83 ¼ �0.1985) and under-esti-mates the order quantity by about 6% ((336.90�357.87)/357.87 ¼ �0.0586).

Table 2 summarises the average productionquantities (APQ), average cost (AC), and percen-tage deviation of average production quantity(PDAPQ) and average cost (PDAC) between SAJMand JBM. These results indicate that the two modelsdiffer the least (PDAC ¼ �0.81%) in terms ofcost, for the case of low cognitive and low motorLRs (fc ¼ 80%, fm ¼ 80%) and R ¼ 1 (corre-sponding to fSAJM ¼ 80%). On the other hand,these models differ the most (PDAC ¼ �19.86%)for the case of high cognitive and high motor LRs(fc ¼ 70%, fm ¼ 90%) and R ¼ 1 (corresponding

ARTICLE IN PRESS

Table 1

Optimal policies for the models with no forgetting of Salameh et al. (SAJM) and Jaber and Bonney (JBM) showing percentage deviations

of order quantities and cost

Cycle Quantity (SAJM) Cost (SAJM) Quantity (JBM) Cost (JBM) % Quantity % Cost

1 462.45 $29.98 473.49 $35.53 �2.33 �15.63

2 328.82 $17.66 372.00 $22.53 �11.61 �21.60

3 325.76 $17.32 346.76 $22.05 �6.06 �21.47

4 324.06 $17.13 343.21 $21.52 �5.58 �20.39

5 322.93 $17.00 342.94 $21.49 �5.84 �20.91

6 322.09 $16.91 342.67 $21.47 �6.01 �21.25

7 321.44 $16.83 339.67 $20.94 �5.37 �19.63

8 320.91 $16.77 339.42 $20.94 �5.45 �19.92

9 320.47 $16.72 339.42 $20.94 �5.58 �20.16

10 320.09 $16.68 339.14 $20.92 �5.62 �20.25

Average 336.90 $18.30 357.87 $22.83 �5.86 �19.85

Table 2

Average production quantities (APQ), average cost (AC), and percentage deviations of these quantities (PDAPQ) and costs (PDAC) for

the models of Salameh et al. (SAJM) and Jaber and Bonney (JBM) for different learning combinations (fc and fm) and R values of the

case of no forgetting

fc (%) fm (%) R APQ (SAJM) AC (SAJM) b (SAJM) (%) APQ (JBM) AC (JBM) PDAPQ (%) PDAC (%)

70 90 1/3 355.07 $21.32 85.00 379.62 $25.69 �6.47 �17.02

70 90 1 336.90 $18.30 80.00 357.87 $22.83 �5.86 �19.86

70 90 3 326.82 $16.95 75.00 343.37 $19.76 �4.82 �14.20

75 85 1/3 344.66 $19.50 82.50 354.61 $20.50 �2.80 �4.88

75 85 1 336.90 $18.30 80.00 343.32 $19.41 �1.87 �5.70

75 85 3 331.13 $17.50 77.50 336.53 $18.29 �1.60 �4.29

80 80 1/3 336.90 $18.30 80.00 338.73 $18.46 �0.54 �0.87

80 80 1 336.90 $18.30 80.00 338.64 $18.45 �0.51 �0.81

80 80 3 336.90 $18.79 80.00 338.82 $18.47 �0.57 �1.73

Where PDAPQ ¼ (APQSAJM�APQJBM)� 100/APQJBM and PDAC ¼ (ACSAJM�ACJBM)� 100/ACJBM.

M.Y. Jaber, M. Bonney / Int. J. Production Economics 108 (2007) 359–367364

to fSAJM ¼ 80%). The results in Table 2 imply thatwhen the task performed is characterised by beinglow cognitive and low motor LRs (fc ¼ 80%,fm ¼ 80%) and for all values of R (correspondingto fSAJM ¼ 80%) the SAJM could be used becauseof the small error (ranging from �0.81 to 1.73).However, if the task performed is highly cognitiveand highly motor, the SAJM is not a goodapproximation.

Now, the above example is reworked usingdifferent learning combinations, R ¼ 1/3, 1, and 3,different interruption times (d ¼ 1, 10, 30, and60 days) and different total forgetting times,characterised by M (M ¼ Dm/Dc). Two cases areconsidered; (1) Dm ¼ 360 and Dc ¼ 180 givingM ¼ 2, and (2) Dm ¼ 360 and Dc ¼ 90 giving

M ¼ 4. These results are summarised in Tables 3and 4, respectively.

Table 3 indicates that the two models differ theleast (PDAC ¼ 1.23%), in terms of cost, for the caseof high cognitive and high motor LRs (fc ¼ 70%,fm ¼ 90%) and R ¼ 3 (corresponding to fSAJM ¼

75%), and d ¼ 60 days. On the other hand, thesemodels differ the most (PDAC ¼ 44.91%) for thecase of medium cognitive and medium motor LRs(fc ¼ 75%, fm ¼ 85%) and R ¼ 1/3 (correspond-ing to fSAJM ¼ 82.5%), and d ¼ 1 days. For thefirst case (fc ¼ 70%, fm ¼ 90%; R ¼ 3), as d

increases from 1 to 30, the PDAC increases from1.68% to 3.46% and decrease to 1.23% as d

increases from 30 to 60, and the correspondingPDAPQ value decreases from 5.75% to �0.92%

ARTICLE IN PRESS

Table 3

The average production quantities (APQ), the average costs (AC), the percentage variations between the average production quantities

(PDAPQ) and costs (PDAC) between the Salameh et al.’s model (SAJM) and Jaber and Bonney’s model (JBM) for different learning

combinations, R and d values, where M ¼ 2

M d fc (%) fm (%) R SAJM JBM

APQ AC b (%) APQ AC PDAPQ (%) PDAC (%)

2 1 70 90 1/3 490.56 $35.78 85.00 398.52 $26.97 23.09 32.66

70 90 1 420.99 $26.57 80.00 367.74 $23.70 14.48 12.11

70 90 3 368.70 $20.65 75.00 348.66 $20.31 5.75 1.68

75 85 1/3 455.61 $30.89 82.50 369.50 $21.32 23.30 44.91

75 85 1 420.99 $26.57 80.00 350.04 $20.07 20.27 32.40

75 85 3 391.49 $23.14 77.50 348.32 $18.78 12.39 23.20

80 80 1/3 421.00 $26.57 80.00 349.29 $19.05 20.53 39.45

80 80 1 421.00 $26.57 80.00 347.25 $19.16 21.24 38.71

80 80 3 421.00 $26.57 80.00 349.69 $19.14 20.39 38.80

2 10 70 90 1/3 498.78 $36.48 85.00 475.68 $28.85 4.86 26.46

70 90 1 444.99 $28.59 80.00 388.18 $25.89 14.64 10.44

70 90 3 395.83 $22.82 75.00 365.80 $22.18 8.21 2.90

75 85 1/3 471.99 $32.27 82.50 437.42 $23.18 7.90 39.23

75 85 1 444.99 $28.59 80.00 404.15 $21.90 10.10 30.56

75 85 3 419.08 $25.42 77.50 377.24 $20.64 11.09 23.15

80 80 1/3 444.99 $28.59 80.00 369.58 $20.99 20.40 36.22

80 80 1 444.99 $28.59 80.00 382.80 $21.05 16.25 35.81

80 80 3 444.99 $28.59 80.00 395.81 $20.96 12.43 36.38

2 30 70 90 1/3 502.19 $36.76 85.00 449.63 $31.80 11.69 15.58

70 90 1 456.45 $29.51 80.00 407.09 $28.13 12.13 4.91

70 90 3 414.36 $24.23 75.00 390.78 $23.42 6.03 3.46

75 85 1/3 478.91 $32.84 82.50 473.13 $25.20 1.22 30.34

75 85 1 456.45 $29.51 80.00 411.19 $23.76 11.01 24.19

75 85 3 434.85 $26.65 77.50 421.00 $22.02 3.29 21.01

80 80 1/3 456.45 $29.51 80.00 415.97 $22.43 9.73 31.59

80 80 1 456.45 $29.51 80.00 420.79 $22.69 8.47 30.05

80 80 3 456.45 $29.51 80.00 426.94 $22.81 6.91 29.35

2 60 70 90 1/3 503.13 $36.84 85.00 476.18 $33.51 5.66 9.94

70 90 1 459.93 $29.78 80.00 444.53 $29.19 3.46 2.03

70 90 3 421.79 $24.77 75.00 425.71 $24.47 �0.92 1.23

75 85 1/3 480.86 $33.00 82.50 428.75 $27.11 12.16 21.74

75 85 1 459.93 $29.78 80.00 433.09 $25.21 6.20 18.11

75 85 3 440.25 $27.07 77.50 394.33 $23.36 11.65 15.89

80 80 1/3 459.93 $29.78 80.00 418.48 $23.81 9.90 25.07

80 80 1 459.93 $29.78 80.00 428.98 $24.10 7.22 23.56

80 80 3 459.93 $29.78 80.00 428.75 $24.20 7.27 23.04

M.Y. Jaber, M. Bonney / Int. J. Production Economics 108 (2007) 359–367 365

with a similar behaviour to PDAC. To investigatethe behaviour of PDAC and PDAPQ, additionalanalyses were conducted for d values (d ¼ 20, 40,and 50). The values of PDAC and PDAPQ wereplotted against the values of d (d ¼ 1, 10, 20, 30, 40,50, and 60). The plots, not shown in this paper forconciseness, indicate that there exists a value of d

that maximises the values of PDAC and PDAPQ,and thus the SAJM may not be a good approxima-tion of JBM for a range of d values. In the secondcase (fc ¼ 75%, fm ¼ 85%; R ¼ 1/3), as d increases

from 1 to 60 the PDAC decreases from 44.91% to21.74%, and the corresponding PDAPQ valuedecreases from 23.30% to 12.16%. As in the earliercase, the values of PDAC and PDAPQ were plottedagainst values of d ¼ 1, 10, 20, 30, 40, 50, and 60, toshow that both PDAC and PDAPQ have adeceasing trend for increasing values of d.

Table 4 indicates that the two models differ the least(PDAC ¼ 0.56%), in terms of cost, for the case of highcognitive and high motor LRs (fc ¼ 70%, fm ¼ 90%)and R ¼ 1 (corresponding to fSAJM ¼ 80%), and

ARTICLE IN PRESS

Table 4

The average production quantities (APQ), the average costs (AC), the percentage variations between the average production quantities

(PDAPQ) and costs (PDAC) between the Salameh et al.’s model (SAJM) and Jaber and Bonney’s model (JBM) for different learning

combinations, R and d values, where M ¼ 4

M d fc (%) fm (%) R SAJM JBM

APQ AC b(%) APQ AC PDAPQ (%) PDAC (%)

4 1 70 90 1/3 490.56 $35.78 85.00 398.70 $26.98 23.04 32.64

70 90 1 420.99 $26.57 80.00 364.92 $23.72 15.36 12.04

70 90 3 368.70 $20.65 75.00 349.47 $20.35 5.50 1.48

75 85 1/3 455.61 $30.89 82.50 369.89 $21.32 23.17 44.92

75 85 1 420.99 $26.57 80.00 351.99 $20.08 19.60 32.36

75 85 3 391.49 $23.14 77.50 344.12 $18.88 13.77 22.60

80 80 1/3 421.00 $26.57 80.00 349.27 $19.10 20.54 39.16

80 80 1 421.00 $26.57 80.00 349.35 $19.24 20.51 38.12

80 80 3 421.00 $26.57 80.00 354.54 $19.25 18.74 38.05

4 10 70 90 1/3 498.78 $36.48 85.00 419.90 $29.49 18.79 23.69

70 90 1 444.99 $28.59 80.00 407.50 $25.92 9.20 10.30

70 90 3 395.83 $22.82 75.00 374.19 $22.41 5.78 1.86

75 85 1/3 471.99 $32.27 82.50 435.91 $23.30 8.28 38.50

75 85 1 444.99 $28.59 80.00 392.22 $22.30 13.45 28.23

75 85 3 419.08 $25.42 77.50 391.51 $20.93 7.04 21.44

80 80 1/3 444.99 $28.59 80.00 370.92 $21.16 19.97 35.12

80 80 1 444.99 $28.59 80.00 403.85 $21.34 10.19 34.00

80 80 3 444.99 $28.59 80.00 403.41 $21.40 10.31 33.57

4 30 70 90 1/3 502.19 $36.76 85.00 446.03 $31.94 12.59 15.07

70 90 1 456.45 $29.51 80.00 417.75 $28.07 9.26 5.15

70 90 3 414.36 $24.23 75.00 424.17 $23.87 �2.31 1.52

75 85 1/3 478.91 $32.84 78.33 473.12 $25.34 1.22 29.62

75 85 1 456.45 $29.51 80.00 398.24 $24.44 14.62 20.73

75 85 3 434.84 $26.65 77.50 443.04 $22.61 �1.85 17.86

80 80 1/3 456.45 $29.51 80.00 416.87 $22.69 9.49 30.06

80 80 1 456.45 $29.51 80.00 449.45 $23.30 1.56 26.66

80 80 3 456.45 $29.51 80.00 443.23 $23.69 2.98 24.58

4 60 70 90 1/3 503.13 $36.84 85.00 476.03 $33.68 5.69 9.39

70 90 1 459.93 $29.78 80.00 444.52 $29.61 3.47 0.56

70 90 3 421.79 $24.77 75.00 425.72 $25.22 �0.92 �1.77

75 85 1/3 480.86 $33.00 82.50 430.86 $27.37 11.61 20.57

75 85 1 459.93 $29.78 80.00 412.53 $26.07 11.49 14.24

75 85 3 440.25 $27.07 77.50 401.33 $24.39 9.70 10.97

80 80 1/3 459.93 $29.78 80.00 418.48 $24.13 9.90 23.41

80 80 1 459.93 $29.78 80.00 428.96 $24.97 7.22 19.25

80 80 3 459.93 $29.78 80.00 445.43 $25.22 3.26 18.06

M.Y. Jaber, M. Bonney / Int. J. Production Economics 108 (2007) 359–367366

d ¼ 60 days. On the other hand, these models differthe most (PDAC ¼ 44.92%) also for the case ofmedium cognitive and medium motor LRs(fc ¼ 75%, fm ¼ 85%) and R ¼ 1/3 (correspond-ing to fSAJM ¼ 82.5%), and d ¼ 1 day. For thefirst case (fc ¼ 70%, fm ¼ 90%; R ¼ 1), as d

increases from 1 to 60 (d ¼ 1, 10, 20, 30, 40, 50,and 60) the PDAC decreases from 12.04% to0.56%, and the corresponding PDAPQ value

decreases in a similar fashion as PDAC from15.36% to 3.47%. In the second case (fc ¼ 75%,fm ¼ 85%; R ¼ 1/3), as d increases from 1 to 60 thePDAC decreases from 44.92% to 20.57%, and thecorresponding PDAPQ value decreases in a similarfashion as PDAC from 23.17% to 11.61%. Theseresults indicate that there may be an advantage inaccounting separately for cognitive and motorelements of a job.

ARTICLE IN PRESSM.Y. Jaber, M. Bonney / Int. J. Production Economics 108 (2007) 359–367 367

6. Summary and conclusions

This paper has studied the effect of lot-sizedependent learning rates on the EMQ problem. Anew lot-size inventory model that accounts for lot-size dependent learning and forgetting rates wasdeveloped. This was compared, for illustration, withthe Jaber and Bonney (1996b) model, whichextended the work of Salameh et al. (1993) thatused invariant learning and forgetting rates. Thisnew model incorporated the dual phase learningand forgetting model (Jaber and Kher, 2002) thataccounts separately for the cognitive and motorelements of a job. The results indicated thatignoring the cognitive and motor structure of atask can result in lot-size policies with highpercentage errors in costs. This suggests that earlierwork that has investigated the lot-size problem inconjunction with learning and forgetting in produc-tion may be unreliable, and therefore should berevisited and possibly revised. This paper hasprovided a framework around which such investiga-tions could be conducted.

Acknowledgement

M.Y. Jaber thanks the Natural Sciences andEngineering Research Council of Canada (NSERC)for supporting his research.

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