Applied Mathematical Modelling 34 (2010) 3623–3630

Contents lists available at ScienceDirect

Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Single-machine scheduling with a general sum-of-actual-processing-times-based and job-position-based learning effect

Yunqiang Yin a,*, Dehua Xu a, Jiayin Wang b

a College of Mathematics and Information Sciences, East China Institute of Technology, Fuzhou, Jiangxi 344000, Chinab School of Mathematical Sciences, Beijing Normal University, Key Laboratory of Mathematics and Complex System, Ministry of Education, Beijing 100875, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 3 December 2008Received in revised form 11 March 2010Accepted 15 March 2010Available online 17 March 2010

Keywords:SchedulingLearning effectSingle-machine

0307-904X/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.apm.2010.03.011

* Corresponding author at: College of Mathematic411 84706402; fax: +86 411 84706405.

E-mail addresses: yinyunqiang@126.com (Y. Yin)

In this paper, we bring into the scheduling field a general learning effect model where theactual processing time of a job is not only a general function of the total actual processingtimes of the jobs already processed, but also a general function of the job’s scheduled posi-tion. We show that the makespan minimization problem and the sum of the kth power ofcompletion times minimization problem can be solved in polynomial time, respectively.We also show that the total weighted completion time minimization problem and themaximum lateness minimization problem can be solved in polynomial time under certainconditions.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

Scheduling problems have received considerable attention since the middle of the last century. However, most researchassumes that job processing times are known and fixed during the whole production process. Recent empirical studies inseveral industries have demonstrated that unit costs decline as companies and employees repeatedly produce a productand gain knowledge or experience. This phenomenon is well-known as a ‘‘learning effect” in the literature [1].

During the last few years, learning effect has attracted growing attention in the scheduling community on account of itssignificance. Therefore, there were many attempts to formulate learning effect in a quantitative form as a function of learningvariables, called a learning curve. Most of the concepts assume that the learning curve is a non-increasing function whichdepends on the jobs already performed. For a survey on learning curves, the reader is referred to Jaber and Bonney [2].

To the best of our knowledge, Biskup [3] and Cheng and Wang [4] were among the pioneers that brought the concept oflearning into the field of scheduling. Biskup introduced a scheduling model with a learning effect in which the actual pro-cessing time of a job is a function of its position in the schedule and showed that the single-machine problems to minimizetotal deviations of job completion times from a common due date and to minimize the sum of job completion times are poly-nomially solvable. Many references have studied such a learning effect model thereafter, a sample of these papers include[4–22].

Note that position-dependent learning effects neglect the processing times of the jobs already processed. If human inter-actions have a significant impact during the processing of jobs, the processing time will add to the employees’ experienceand cause learning effects. For situations like this it might be more appropriate to consider a time-dependent learning effect.Kuo and Yang [23] considered a single-machine scheduling problem with a time-dependent learning effect. The time-depen-dent learning effect of a job is assumed to be a function of the total normal processing time of the jobs scheduled in front of

. All rights reserved.

s and Information Sciences, East China Institute of Technology, Fuzhou, Jiangxi 344000, China. Tel.: +86

, schedulingresearch@gmail.com (D. Xu), wjy@bnu.edu.cn (J. Wang).

3624 Y. Yin et al. / Applied Mathematical Modelling 34 (2010) 3623–3630

it. They showed that the SPT sequence is the optimal sequence for the objective of minimizing the total completion time. Formore papers about this time-dependent learning model, the reader is referred to [24–28]. Besides, Koulamas and Kyparisis[29] introduced a general sum-of-job-processing- times-based learning effect scheduling model in which employees learnmore if they perform a job with a longer processing time and showed that the two-machine flowshop makespan and totalcompletion time minimization problems are solvable by the SPT sequencing rule when the job processing times are orderedand job-position-based learning is in effect, and that the flowshop results apply also in the more general sum-of-job-processing-times-based learning environment when the more specialized proportional job processing times are in place.

Wu and Lee [30] proposed a new learning model where the actual job processing time not only depends on its scheduledposition, but also depends on the sum of the processing times of jobs already processed and showed that the single-machinemakespan and the total completion time problems remain polynomially solvable under the proposed model and that thetotal weighted completion time has a polynomial optimal solution under certain agreeable conditions. Cheng et al. [31] fur-ther consider a similar model. They obtained similar results as in Wu and Lee [30], see also [32]. Yin et al. [33] provided amore general model with learning effects where the actual processing time of a job is not only a function of the total normalprocessing times of the jobs already processed, but also a function of the job’s scheduled position, which is a significantextension of some of the existing results on learning effects in the literature. In particular, it is shown that some single ma-chine scheduling problems and m-machine permutation flowshop problems are still polynomially solvable under the pro-posed model.

For a recent state-of-the-art literature review on scheduling problems with learning effect considerations, the reader isreferred to Biskup [34].

Recently, Yang and Kuo [35] pointed out that it is more appropriate to express the learning as a function of the total actualprocessing time of the already processed jobs in many situations and proposed a new time-dependent learning effect modelin which the actual processing time of a job is affected by the total actual processing time of the previous jobs’ schedules. Inthis paper we develop a general learning effect model where the actual processing time of a job is not only a general functionof the total actual processing times of the jobs already processed, but also a general function of the job’s scheduled position.The remainder of this paper is organized as follows. In Section 2, we formulate the general learning effect model. In Section 3,we consider the single-machine problems to minimize makespan, sum of the kth power of completion times, total weightedcompletion time, and maximum lateness under the general learning effect model. We conclude the paper in the last section.

2. A general learning effect model

Assume that there are n jobs J1; J2; . . . ; Jn to be processed on a single-machine. The machine can handle one job at a timeand preemption is not allowed. Each job Jj has a normal processing time pj and a due date dj. In addition, let p½k� and pA

½k� be thenormal processing time and the actual processing time of a job when it is scheduled in position k in a sequence, respectively.If job Ji is scheduled in position r in a sequence, then its actual processing time is defined as

pAir ¼ pif

Pr�1k¼1pA

½k�Pnk¼1pk

!gðrÞ; ð1Þ

whereP0

k¼1pA½k� ¼ 0; pA

½1� ¼ p½1�; pA½s� ¼ p½s�f

Ps�1

k¼1pA½k�Pn

k¼1pk

� �gðrÞð1 < s 6 r � 1Þ; f is a differentiable non-increasing function from

½0;þ1Þ into (0,1] with f 0 is non-decreasing on ½0;þ1Þ; f ð0Þ ¼ 1 and �1 6 f 0ðxÞ 6 0, and g is a non-increasing function from½1;þ1Þ into (0,1] with gð1Þ ¼ 1. For convenience, we denote the learning effect given in Eq. (1) by LEgatp. In such a learningmodel, we can see that the actual processing time of a job is not only a general function of the total actual processing times ofthe jobs already processed, but also a general function of the job’s scheduled position. It is also evident from the model thatthe longer the already processed jobs or the later a job is scheduled, the steeper the learning effects for subsequent jobs notprocessed yet.

For a given schedule S, let CjðSÞ denote the completion time of job Jj and C½r�ðSÞ represent the complete time of the jobscheduled in position r in S. Then the completion time of the rth job in S under Eq. (1) is

C½r�ðSÞ ¼Xr

k¼1

p½k�f

Pk�1l¼1 pA

½l�Pnl¼1pl

!gðkÞ: ð2Þ

Remark 2.1. If f is defined as f ðxÞ ¼ 1 for all x 2 ½0;þ1Þ. Then model (2) is a generalization of the model proposed in Biscup[3]. Moreover, if g is given by gðxÞ ¼ 1 for all x 2 ½1;þ1Þ, then the learning effect depends only on the actual processingtimes of the jobs already scheduled.

3. Some single-machine scheduling problems

In the classical single-machine makespan minimizing problem, the makespan value is sequence-independent. However,this may be different when the learning effect is considered. Here, we study the makespan minimization problem on a

Y. Yin et al. / Applied Mathematical Modelling 34 (2010) 3623–3630 3625

single-machine under the general learning effect model, denoted by 1jLEgatpjCmax. And we show that the optimal solution forthe problem can be obtained by the shortest processing time (SPT) rule. Before presenting the main result, we first introducea useful lemma.

Lemma 3.1. If f is a differentiable non-increasing function from ½0;þ1Þ into (0,1] satisfying f 0 is non-decreasing and g is a non-increasing function from ½1;þ1Þ into (0,1], then

ð1� aÞf ðaÞ þ af ðaþ tÞ gðr þ 1ÞgðrÞ � f ðaþ atÞ gðr þ 1Þ

gðrÞ 6 0;

for a P 1; t P 0 and r ¼ 1;2; . . . ;n.

Proof. Let

FðtÞ ¼ ð1� aÞf ðaÞ þ af ðaþ tÞ gðr þ 1ÞgðrÞ � f ðaþ atÞ gðr þ 1Þ

gðrÞ :

Taking the first derivative of f ðtÞ with respect to t, we have

F 0ðtÞ ¼ af 0ðaþ tÞ gðr þ 1ÞgðrÞ � af 0ðaþ atÞ gðr þ 1Þ

gðrÞ ¼ aðf 0ðaþ tÞ � f 0ðaþ atÞÞ gðr þ 1ÞgðrÞ :

Since a P 1; t P 0 and f 0 is non-decreasing, we have f 0ðaþ tÞ � f 0ðaþ atÞ 6 0 and so F 0ðtÞ 6 0. This implies that FðtÞ is non-increasing on t P 0. Since a P 1; f ðaÞP 0 and g is non-increasing, we have

FðtÞ 6 Fð0Þ ¼ ð1� aÞf ðaÞ þ af ðaÞ gðr þ 1ÞgðrÞ � f ðaÞ gðr þ 1Þ

gðrÞ ¼ ð1� aÞf ðaÞ 1� gðr þ 1ÞgðrÞ

� �6 0:

This completes the proof. h

Theorem 3.2. For the problem 1jLEgatpjCmax, there exists an optimal schedule in which the jobs are ordered according to the SPTrule.

Proof. By pairwise job interchange technique. Assume that there exists an optimal schedule S ¼ ðp1; Jj; Ji;p2Þ with pj > pi,where p1 and p2 denote the partial sequences of S. Let S0 be a schedule with jobs Ji and Jj of S mutually exchanged, thatis, S0 ¼ ðp1; Ji; Jj;p2Þ. Furthermore, we assume that there are r � 1 jobs in p1. We will show that the interchange of jobs Ji

and Jj does not increase the objective value. Repeating this interchanging argument will lead to the optimality of the SPTsequence for the problem 1jLEgatpjCmax. Specifically, it suffices to show that CjðS0Þ 6 CiðSÞ and ClðS0Þ 6 ClðSÞ for any job Jl inp2. To further simplify the notation, let B denote the completion time of the last job in p1 and P ¼

Pnk¼1pk. From Eq. (2),

the completion times of Jj in S0 and Ji in S are respectively

CjðS0Þ ¼ Bþ pifBP

� �gðrÞ þ pjf

Bþ pifBP

� �gðrÞ

P

� �gðr þ 1Þ

and

CiðSÞ ¼ Bþ pjfBP

� �gðrÞ þ pif

Bþ pjfBP

� �gðrÞ

P

!gðr þ 1Þ:

Then we have

CjðS0Þ � CiðSÞ ¼ pifBP

� �gðrÞ þ pjf

Bþ pifBP

� �gðrÞ

P

� �gðr þ 1Þ � pjf

BP

� �gðrÞ � pif

Bþ pjfBP

� �gðrÞ

P

!gðr þ 1Þ

¼ pigðrÞ 1�pj

pi

� �f

BP

� �þ

pj

pif

Bþ pifBP

� �gðrÞ

P

� �gðr þ 1Þ

gðrÞ � fBþ pjf

BP

� �gðrÞ

P

!gðr þ 1Þ

gðrÞ

!:

Let a ¼ pj

pi. Clearly, a P 1 and so � � � �

CjðS0Þ � CiðSÞ ¼ pigðrÞ ð1� aÞf BP

� �þ af

BPþ

pifBP gðrÞP

� �gðr þ 1Þ

gðrÞ � fBPþ a

pifBP gðrÞP

� �gðr þ 1Þ

gðrÞ

� �6 0 ðBy Lemma3:1Þ:

This implies that CjðS0Þ 6 CiðSÞ. Now let Jk be the first job scheduled in p2. Then from Eq. (2) the completion times of the job Jk

in S0 and S are given by

CkðS0Þ ¼ CjðS0Þ þ pkfCjðS0Þ

P

� �gðr þ 2Þ

3626 Y. Yin et al. / Applied Mathematical Modelling 34 (2010) 3623–3630

and

CkðSÞ ¼ CiðSÞ þ pkfCiðSÞ

P

� �gðr þ 2Þ;

respectively. By Lagrange mean value theorem, there exists a point n between CjðS0Þ and CiðSÞ such that

fCjðS0Þ

P

� �� f

CiðSÞP

� �¼ f 0

nP

� �CjðS0Þ � CiðSÞ

P:

Thus we have

CkðS0Þ � CkðSÞ ¼ ðCjðS0Þ � CiðSÞÞ þ pkgðr þ 2Þ fCjðS0Þ

P

� �� f

CiðSÞP

� �� �¼ ðCjðS0Þ � CiðSÞÞ þ pkgðr þ 2Þf 0 n

P

� �CjðS0Þ � CiðSÞ

P

¼ ðCjðS0Þ � CiðSÞÞ 1þ pkgðr þ 2ÞP

f 0nP

� �� �:

Since 0 < pk < P; 0 < gðr þ 2Þ < 1 and �1 6 f 0 nP

� �6 0, we have 1þ pkgðrþ2Þ

P f 0 nP

� �> 0. Hence CjðS0Þ 6 CiðSÞ implies CkðS0Þ 6 CkðSÞ.

In a similar way, we have ClðS0Þ 6 ClðSÞ for any job Jl in p2. This completes the proof. h

Note that Theorem 3.1 does not hold in general if f 0ðxÞP 1 for some x 2 ½0;þ1Þ as shown in the following example.

Example 3.3. Let f ðxÞ ¼ ð1þ xÞ�2 and gðxÞ ¼ x�2. Given n ¼ 3; p1 ¼ 1; p2 ¼ 2 and p3 ¼ 100. The SPT sequence ðJ1; J2; J3Þyields a makespan of 3.5856, while the sequence ðJ2; J1; J3Þ yields the optimal makespan of 3.2398.

Townsend [36] studied a single-machine scheduling problem with a quadratic cost function of completion times andshowed that the problem 1k

PC2

j can be solved optimally by the SPT rule. In some scheduling situations, it is possible toconsider a polynomial cost function of degree k. Therefore, we study the sum of the kth power of completion times on a sin-gle-machine under the general learning effect model, denoted by 1jLEgatpj

PCk

j . And we show that the SPT rule is still optimalfor the problem 1jLEgatpj

PCk

j . The result is stated in the following theorem.

Theorem 3.4. For the problem 1jLEgatpjP

Ckj , where k is a positive real number, there exists an optimal schedule in which the jobs

are ordered according to the SPT rule.

Proof. We adopt the same notations as in the proof of Theorem 3.2. From Eq. (2), the completion times of Ji in S0 and Jj in Sare respectively

CiðS0Þ ¼ Bþ pifBP

� �gðrÞ

and

CjðSÞ ¼ Bþ pjfBP

� �gðrÞ:

Since pi < pj, it follows that CiðS0Þ 6 CjðSÞ. By the proof of Theorem 3.2, we have CjðS0Þ 6 CiðSÞ and ClðS0Þ 6 ClðSÞ for any jobJlðl – i; jÞ. Thus we have C½r�ðS0Þ 6 C½r�ðSÞ for 1 6 r 6 n. Since k is a positive real number, we haveP

Ckj ðS0Þ ¼

PCk½r�ðS

0Þ 6P

Ck½r�ðSÞ ¼

PCk

j ðSÞ. This completes the proof. h

From Theorem 3.4, we obtain immediately the following corollary.

Corollary 3.5. For total completion time minimization problem 1jLEgatpjP

Cj, there exists an optimal schedule in which the jobsare ordered according to the SPT rule.

For the objective functions of minimizing the weighted sum of completion times and minimizing maximum lateness,Mosheiov [13] showed that polynomial optimal solutions of the classical version do not hold with the learning effect. How-ever, under certain conditions the problems can be solved in polynomial time under the general learning effect model. First,we give two useful lemmas for the following theorem.

Lemma 3.6. If f is a differentiable non-increasing function from ½0;þ1Þ into (0,1] satisfying f 0 is non-decreasing and g is a non-increasing function from ½1;þ1Þ into (0,1], then

�f ðaÞ þ k1f ðaþ tÞ gðr þ 1ÞgðrÞ � k2tf 0ðaþ tÞ gðr þ 1Þ

gðrÞ 6 0

for 0 6 k1 6 k2 6 1; t P 0; and r ¼ 1;2; . . . ;n:

Y. Yin et al. / Applied Mathematical Modelling 34 (2010) 3623–3630 3627

Proof. Let

Fðk1; k2; tÞ ¼ �f ðaÞ þ k1f ðaþ tÞ gðr þ 1ÞgðrÞ � k2tf 0ðaþ tÞ gðr þ 1Þ

gðrÞ :

Then @F@k1¼ f ðaþ tÞ gðrþ1Þ

gðrÞ P 0; @F@k2¼ �tf 0ðaþ tÞ gðrþ1Þ

gðrÞ P 0. Since 0 6 k1 6 k2 6 1, we have

Fðk1; k2; tÞ 6 Fð1;1; tÞ ¼ �f ðaÞ þ f ðaþ tÞ gðr þ 1ÞgðrÞ � tf 0ðaþ tÞ gðr þ 1Þ

gðrÞ :

By Lagrange mean value theorem, there exists a point n between a and aþ t such that

f ðaþ tÞ � f ðaÞ ¼ f 0ðnÞt:

Since t P 0; f 0 is non-decreasing and g is non-increasing, we have

Fðk1; k2; tÞ 6 �f ðaÞ þ f ðaþ tÞ gðr þ 1ÞgðrÞ � tf 0ðaþ tÞ gðr þ 1Þ

gðrÞ 6 �f ðaÞ gðr þ 1ÞgðrÞ þ f ðaþ tÞ gðr þ 1Þ

gðrÞ � tf 0ðaþ tÞ gðr þ 1ÞgðrÞ

¼ tðf 0ðnÞ � f 0ðaþ tÞÞ gðr þ 1ÞgðrÞ 6 0:

This completes the proof. h

Lemma 3.7. If f is a differentiable non-increasing function from ½0;þ1Þ into (0,1] satisfying f 0 is non-decreasing and g is a non-increasing function from ½1;þ1Þ into ð0;1�, then

ð1� aÞf ðaÞ þ ak1f ðaþ tÞ gðr þ 1ÞgðrÞ � k2f ðaþ atÞ gðr þ 1Þ

gðrÞ 6 0

for a P 1; t P 0; 0 6 k1 6 k2 6 1 and r ¼ 1;2; . . . ;n:

Proof. Let

FðaÞ ¼ ð1� aÞf ðaÞ þ ak1f ðaþ tÞ gðr þ 1ÞgðrÞ � k2f ðaþ atÞ gðr þ 1Þ

gðrÞ :

Taking the first derivative of FðaÞ with respect to a, we have

F 0ðaÞ ¼ �f ðaÞ þ k1f ðaþ tÞ gðr þ 1ÞgðrÞ � k2tf 0ðaþ atÞ gðr þ 1Þ

gðrÞ :

Since a P 1; t P 0 and f 0 is non-decreasing, by Lemma 3.6, we have

F 0ðaÞ 6 F 0ð1Þ ¼ �f ðaÞ þ k1f ðaþ tÞ gðr þ 1ÞgðrÞ � k2tf 0ðaþ tÞ gðr þ 1Þ

gðrÞ 6 0:

This implies that FðaÞ is non-increasing on a P 1 and so

FðaÞ 6 Fð1Þ ¼ k1f ðaþ tÞ gðr þ 1ÞgðrÞ � k2f ðaþ tÞ gðr þ 1Þ

gðrÞ ¼ f ðaþ tÞðk1 � k2Þgðr þ 1Þ

gðrÞ 6 0:

This completes the proof. h

Theorem 3.8. For the problem 1jLEgatpjP

wjCj, if jobs have reversely agreeable weights, that is, pi 6 pj implies wi P wj for all jobsJi and Jj, then there exists an optimal schedule in which the jobs are ordered in non-decreasing order of pj

wj(weighted shortest pro-

cessing time rule, WSPT rule).

Proof. By pairwise job interchange argument. We still adopt the same notations as in the proof of Theorem 3.2. Assume thatthere is an optimal schedule which does not follow the WSPT rule, that is, there exist at least two adjacent jobs, say job Jj

scheduled before Ji with pj

wj> pi

wi. Note that the weights of the two jobs are reversely agreeable by the assumption, we have

pj P pi and wj 6 wi. Next we will show that the interchange of jobs Ji and Jj does not increase objective value. Repeating thisinterchanging argument will lead to the optimality of the WSPT sequence for the problem 1jLEgatpj

PwjCj. In particular, it

suffices to show that wiCiðS0Þ þwjCjðS0Þ 6 wiCiðSÞ þwjCjðSÞ since C½l�ðS0Þ 6 C½l�ðSÞ for 1 6 l 6 n by the proof of Theorem 3.2.From Eq. (2), we have

wiCiðS0Þ þwjCjðS0Þ ¼ wjBþwjpifBP

� �gðrÞ þwjpjf

Bþ pifBP

� �gðrÞ

P

� �gðr þ 1Þ þwiBþwipif

BP

� �gðrÞ

3628 Y. Yin et al. / Applied Mathematical Modelling 34 (2010) 3623–3630

and

wiCiðSÞ þwjCjðSÞ ¼ wiBþwipjfBP

� �gðrÞ þwipif

Bþ pjfBP

� �gðrÞ

P

!gðr þ 1Þ þwjBþwjpjf

BP

� �gðrÞ:

Then we have

wiCiðS0Þ þwjCjðS0Þ �wiCiðSÞ �wjCjðSÞ

¼ ðwi þwjÞðpi � pjÞfBP

� �gðrÞ þwjpjf

Bþ pifBP

� �gðrÞ

P

� �gðr þ 1Þ �wipif

Bþ pjfBP

� �gðrÞ

P

!gðr þ 1Þ

¼ ðwi þwjÞpigðrÞ 1�pj

pi

� �f

BP

� �þ wj

wi þwj

pj

pif

BPþ

pifBP

� �gðrÞ

P

� �gðr þ 1Þ

gðrÞ � wi

wi þwjf

BPþ

pjfBP

� �gðrÞ

P

!gðr þ 1Þ

gðrÞ

!:

Let a ¼ pj

pi; k1 ¼

wj

wiþwjand k2 ¼ wi

wiþwj. Clearly a > 1 and 0 6 k1 6 k2 6 1. Thus, by Lemma 3.7, we have

wiCiðS0Þ þwjCjðS0Þ �wiCiðSÞ �wjCjðSÞ

¼ ðwi þwjÞpigðrÞ ð1� aÞf BP

� �þ k1af

BPþ

pifBP

� �gðrÞ

P

� �gðr þ 1Þ

gðrÞ � k2fBPþ a

pifBP

� �gðrÞ

P

� �gðr þ 1Þ

gðrÞ

� �6 0:

Consequently, wiCiðS0Þ þwjCjðS0Þ 6 wiCiðSÞ þwjCjðSÞ. This completes the proof. h

If the normal processing times of all jobs are equal, that is, pj ¼ p for 1 6 j 6 n, then we have the following corollary.

Corollary 3.9. For the problem 1jLEgatp; j ¼ pjP

wjCj, there exists an optimal schedule in which the jobs are ordered in non-increasing order of wj.

Let k be a non-negative real number. If wj ¼ kpj

, then jobs have reversely agreeable weights, that is, pi 6 pj implies wi P wj

for all jobs Ji and Jj. Hence, we have the following corollary.

Corollary 3.10. For the problem 1jLEgatp; j ¼ kpjjP

wjCj, where k is a non-negative real number, there exists an optimal schedulein which the jobs are ordered according to the SPT rule.

Next, we will show that the single-machine problem to minimize maximum lateness under the general learning effectmodel, denoted by 1jLEgatpjLmax, can be solved in polynomial time if the job processing times and the due dates are agreeable,that is, di 6 dj implies pi 6 pj for all jobs Ji and Jj.

Theorem 3.11. For the problem 1jLEgatpjLmax, if the job processing times and the due dates are agreeable, that is, di 6 dj impliespi 6 pj for all jobs Ji and Jj, then there exists an optimal schedule in which the jobs are ordered in non-decreasing order of dj (that is,Earliest due date rule, EDD rule).

Proof. By pairwise job interchange argument. We still adopt the same notations as in the proof of Theorem 3.2. Assume thatthere is an optimal schedule which does not follow the EDD rule, that is, there exist at least two adjacent jobs, say job Ji andjob Jj such that Jj is scheduled before Ji with dj > di, which implies pj P pi by the assumption. Next we will show that themaximum lateness does not increase by interchanging the jobs Ji and Jj. Repeating this interchanging argument will leadto the optimality of the EDD sequence for the problem 1jLEgatpjLmax. Specially, it is suffices to show that maxfLiðS0Þ;LjðS0Þg 6maxfLiðSÞ; LjðSÞg since C½l�ðS0Þ 6 C½l�ðSÞ for 1 6 l 6 n by the proof of Theorem 3.2. By definition, the lateness of jobsJi and Jj in S and jobs Jj and Ji in S0 are respectively

LiðSÞ ¼ CiðSÞ � di;

LjðSÞ ¼ CjðSÞ � dj;

LiðS0Þ ¼ CiðS0Þ � di;

and

LjðS0Þ ¼ CjðS0Þ � dj:

Since dj P di and CjðS0Þ 6 CiðSÞ, we have LjðS0Þ 6 LiðSÞ. In addition, since job Ji is processed before job Jj in S0, we haveCiðS0Þ 6 CjðS0Þ 6 CiðSÞ and so LiðS0Þ 6 LiðSÞ. Therefore, we have

maxfLiðS0Þ; LjðS0Þg 6 maxfLiðSÞ; LjðSÞg:

This completes the proof. h

If the normal processing times of all jobs are equal, that is, pj ¼ p for 1 6 j 6 n, then we have the following corollary.

Corollary 3.12. For the problem 1jLEgatp; pj ¼ pjLmax, there exists an optimal schedule in which the jobs are ordered according tothe EDD rule.

Y. Yin et al. / Applied Mathematical Modelling 34 (2010) 3623–3630 3629

Let k be a non-negative real number. If dj ¼ kpj, then the job processing times and the due dates are agreeable, that is,di 6 dj implies pi 6 pj for all the jobs Ji and Jj. Hence, we have the following corollary.

Corollary 3.13. For the problem 1jLEgatp; dj ¼ kpjjLmax, where k is a non-negative real number, there exists an optimal schedule inwhich the jobs are ordered according to the EDD rule.

If all jobs share a common due date, that is, dj ¼ d for 1 6 j 6 n, then we have the following corollary.

Corollary 3.14. For the problem 1jLEgatp; dj ¼ djLmax, there exists an optimal schedule in which the jobs are ordered according tothe SPT rule.

4. Conclusions

Scheduling problems with a learning effect have recently received growing attention from the scheduling research com-munity. The aim of the paper is to bring into the scheduling field a general learning effect model where the actual processingtime of a job is not only a general function of the total actual processing times of the jobs already processed, but also a gen-eral function of the job’s scheduled position. Particularly, we showed that the single-machine makespan and the sum of thekth power of completion times problems remain polynomially solvable under the general learning effect model. In addition,we showed that the total weighted completion time and the maximum lateness problems have polynomial optimal solutionsunder certain conditions. We believe that the general learning effect model offered here will turn out to be more useful in thetheory and applications of scheduling.

Acknowledgements

We express our warmest thanks to the referees for their interest in our work and their valuable comments for improvingthe paper. This research is sponsored by priority discipline of Beijing Normal University.

References

[1] A.B. Badiru, Computational survey of univariate and multivariate learning curve models, IEEE Trans. Eng. Manage. 39 (1992) 176–188.[2] Y.M. Jaber, M. Bonney, The economic manufacture/order quantity (EMQ/EOQ) and the learning curve: past, present, and future, Int. J. Prod. Econ. 59

(1999) 93–102.[3] D. Biskup, Single-machine scheduling with learning considerations, Eur. J. Oper. Res. 115 (1999) 173–178.[4] T.C.E. Cheng, G. Wang, Single-machine scheduling with learning effect considerations, Ann. Oper. Res. 98 (2000) 273–290.[5] A. Bachman, A. Janiak, Scheduling jobs with position-dependent processing times, J. Oper. Res. Soc. (2004) 257–263.[6] P. Chen, C.-C. Wu, W.-C. Lee, A bi-criteria two-machine flowshop scheduling problem with a learning effect, J. Oper. Res. Soc. 57 (2006) 1113–1125.[7] M.B. Cheng, S.J. Sun, Y. Yu, A note on flow shop scheduling problems with a learning effect on no-idle dominant machines, Appl. Math. Comput. 184

(2007) 945–949.[8] T. Eren, E. Guner, Minimizing total tardiness in a scheduling problem with a learning effect, Appl. Math. Model. 31 (2007) 1351–1361.[9] W.-C. Lee, C.-C. Wu, Minimizing total completion time in a two-machine flowshop with a learning effect, Int. J. Prod. Econ. 88 (2004) 85–92.

[10] W.-C. Lee, C.-C. Wu, A note on single-machine group scheduling problems with position-based learning effect, Appl. Math. Model. 33 (4) (2009) 2159–2163.

[11] W.-C. Lee, C.-C. Wu, H.-J. Sung, A bi-criterion single-machine scheduling problem with learning considerations, Acta Inform. 40 (2004) 303–315.[12] B.M.T. Lin, Complexity results for single-machine scheduling with positional learning effects, J. Oper. Res. Soc. 58 (2007) 1099–1102.[13] G. Mosheiov, Scheduling problems with a learning effect, Eur. J. Oper. Res. 132 (2001) 687–692.[14] G. Mosheiov, Parallel machine scheduling with a learning effect, J. Oper. Res. Soc. 52 (2001) 1165–1169.[15] G. Mosheiov, J.B. Sidney, Scheduling with general job-dependent learning curves, Eur. J. Oper. Res. 147 (2003) 665–670.[16] J.-B. Wang, A note on scheduling problems with learning effects and deteriorating jobs, Int. J. Syst. Sci. (2006) 827–832.[17] J.-B. Wang, Single-machine scheduling problems with the effects of learning and deterioration, Omega 35 (2007) 397–402.[18] X. Wang, T.C.E. Cheng, Single-machine scheduling with deteriorating jobs and learning effects to minimize the makespan, Eur. J. Oper. Res. 178 (2007)

57–70.[19] J.-B. Wang, Z.-Q. Xia, Flow-shop scheduling with a learning effect, J. Oper. Res. Soc. 56 (2005) 1325–1330.[20] C.-C. Wu, W.-C. Lee, T. Chen, Heuristic algorithms for solving the maximum lateness scheduling problem with learning considerations, Comput. Ind.

Eng. 52 (2007) 124–132.[21] C.-C. Wu, W.-C. Lee, W.-C. Wang, A two-machine flowshop maximum tardiness scheduling problem with a learning effect, Int. J. Adv. Manuf. Technol.

31 (2007) 743–750.[22] C.-L. Zhao, Q.-L. Zhang, H.-Y. Tang, Machine scheduling problems with learning effects, dynamics of continuous, discrete and impulsive systems, Ser. A:

Math. Anal. 11 (2004) 741–750.[23] W.-H. Kuo, D.-L. Yang, Minimizing the total completion time in a single-machine scheduling problem with a time-dependent learning effect, Eur. J.

Oper. Res. 174 (2006) 1184–1190.[24] W.-H. Kuo, D.-L. Yang, Single-machine group scheduling with a time-dependent learning effect, Comput. Oper. Res. 33 (2006) 2099–2112.[25] W.-H. Kuo, D.-L. Yang, Single-machine scheduling problems with the time-dependent learning effect, Comput. Math. Appl. 53 (2007) 1733–1739.[26] W.-C. Lee, C.-C. Wu, P.-H. Hsu, A single-machine learning effect scheduling problem with release times, Omega 38 (2010) 3–11.[27] J.-B. Wang, C.T. Ng, T.C.E. Cheng, L.L. Liu, Single-machine scheduling with a time-dependent learning effect, Int. J. Prod. Econ. 111 (2008) 802–811.[28] C.-C. Wu, W.-C. Lee, A note on the total completion time problem in a permutation flowshop with a learning effect, Eur. J. Oper. Res. 192 (1) (2009)

343–347.[29] C. Koulamas, G.J. Kyparisis, Single-machine and two-machine flowshop scheduling with general learning functions, Eur. J. Oper. Res. 178 (2007) 402–

407.[30] C.-C. Wu, W.-C. Lee, Single-machine scheduling problems with a learning effect, Appl. Math. Model. 32 (2008) 1191–1197.

3630 Y. Yin et al. / Applied Mathematical Modelling 34 (2010) 3623–3630

[31] T.C.E. Cheng, C.-C. Wu, W.-C. Lee, Some scheduling problems with sum of-processing times-based and job-position-based learning effects, Inf. Sci. 178(2008) 2476–2487.

[32] C.-C. Wu, W.-C. Lee, Single-machine and flowshop scheduling with a general learning effect model, Comput. Ind. Eng. 56 (2009) 1553–1558.[33] Y. Yin, D. Xu, K. Sun, H. Li, Some scheduling problems with general position-dependent and time-dependent learning effects, Inf. Sci. 179 (2009) 2416–

2425.[34] D. Biskup, A state-of-the-art review on scheduling with learning effects, Eur. J. Oper. Res. 188 (2008) 315–329.[35] D.-L. Yang, W.-H. Kuo, Single-machine scheduling with an actual time-dependent learning effect, J. Oper. Res. Soc. 58 (2007) 1348–1353.[36] W. Townsend, The single-machine problem with quadratic penalty function of completion times: a branch-and-bound solution, Manage. Sci. 24 (1978)

530–533.