9
Notes on ‘‘some single-machine scheduling problems with general position-dependent and time-dependent learning effects’’ Yunqiang Yin a , Dehua Xu a,, Xiaokun Huang b a School of Mathematics and Information Sciences, East China Institute of Technology, Fuzhou, Jiangxi 344000, China b Department of Mathematics, Honghe University, Mengzi, Yunnan 661100, China article info Article history: Received 5 October 2009 Received in revised form 28 November 2010 Accepted 11 January 2011 Available online 18 January 2011 Keywords: Scheduling Learning effect Single machine abstract This paper provides a continuation of the idea presented by Yin et al. [Yin et al., Some scheduling problems with general position-dependent and time-dependent learning effects, Inform. Sci. 179 (2009) 2416–2425]. For each of the following three objectives, total weighted completion time, maximum lateness and discounted total weighted completion time, this paper presents an approximation algorithm which is based on the optimal algo- rithm for the corresponding single-machine scheduling problem and analyzes its worst- case bound. It shows that the single-machine scheduling problems under the proposed model can be solved in polynomial time if the objective is to minimize the total lateness or minimize the sum of earliness penalties. It also shows that the problems of minimizing the total tardiness, discounted total weighted completion time and total weighted earliness penalty are polynomially solvable under some agreeable conditions on the problem parameters. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction Scheduling problems have received considerable attention since the middle of the last century. However, most research assumes that job processing times are known and fixed during the whole production process [16,19,24–27]. Recent empirical studies in several industries have demonstrated that unit costs decline as companies produce more of a product and gain knowledge or experience. This phenomenon is well-known as the ‘‘learning effect’’ in the literature [1,2]. To the best of our knowledge, Biskup [2] and Cheng and Wang [5] were among the pioneers that brought the concept of learning effect into the field of scheduling. Biskup [2] introduced a scheduling model with position-dependent learning ef- fects in which the actual processing time p jr of job J j when it is scheduled in position r in a processing sequence is defined as p jr ¼ p j r a ; where p j is the normal processing time of job J j and a (a < 0) is the learning index. He showed that the single-machine sched- uling problems under the proposed model are polynomially solvable if the objective is to minimize the deviation from a common due date or to minimize the sum of flow times. Mosheiov [15] investigated several other single-machine scheduling problems, and the problem of minimizing the total flow time on identical parallel machines. Lee et al. [12] considered a bicri- terion single-machine scheduling problem under the above learning effect model. The objective is to find a sequence that minimizes a linear combination of the total completion time and the maximum tardiness. Liu et al. [13] proved that the weighted shortest processing time (WSPT) rule, the earliest due date (EDD) rule and the modified Moore–Hodgson algorithm 0020-0255/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2011.01.018 Corresponding author. Tel./fax: +86 794 8258307. E-mail addresses: [email protected] (Y. Yin), [email protected] (D. Xu). Information Sciences 181 (2011) 2209–2217 Contents lists available at ScienceDirect Information Sciences journal homepage: www.elsevier.com/locate/ins

Notes on “some single-machine scheduling problems with general position-dependent and time-dependent learning effects”

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Page 1: Notes on “some single-machine scheduling problems with general position-dependent and time-dependent learning effects”

Information Sciences 181 (2011) 2209–2217

Contents lists available at ScienceDirect

Information Sciences

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

Notes on ‘‘some single-machine scheduling problems with generalposition-dependent and time-dependent learning effects’’

Yunqiang Yin a, Dehua Xu a,⇑, Xiaokun Huang b

a School of Mathematics and Information Sciences, East China Institute of Technology, Fuzhou, Jiangxi 344000, Chinab Department of Mathematics, Honghe University, Mengzi, Yunnan 661100, China

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

Article history:Received 5 October 2009Received in revised form 28 November 2010Accepted 11 January 2011Available online 18 January 2011

Keywords:SchedulingLearning effectSingle machine

0020-0255/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.ins.2011.01.018

⇑ Corresponding author. Tel./fax: +86 794 825830E-mail addresses: [email protected] (Y. Yi

This paper provides a continuation of the idea presented by Yin et al. [Yin et al., Somescheduling problems with general position-dependent and time-dependent learningeffects, Inform. Sci. 179 (2009) 2416–2425]. For each of the following three objectives, totalweighted completion time, maximum lateness and discounted total weighted completiontime, this paper presents an approximation algorithm which is based on the optimal algo-rithm for the corresponding single-machine scheduling problem and analyzes its worst-case bound. It shows that the single-machine scheduling problems under the proposedmodel can be solved in polynomial time if the objective is to minimize the total latenessor minimize the sum of earliness penalties. It also shows that the problems of minimizingthe total tardiness, discounted total weighted completion time and total weighted earlinesspenalty are polynomially solvable under some agreeable conditions on the problemparameters.

� 2011 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 [16,19,24–27]. Recent empiricalstudies in several industries have demonstrated that unit costs decline as companies produce more of a product and gainknowledge or experience. This phenomenon is well-known as the ‘‘learning effect’’ in the literature [1,2].

To the best of our knowledge, Biskup [2] and Cheng and Wang [5] were among the pioneers that brought the concept oflearning effect into the field of scheduling. Biskup [2] introduced a scheduling model with position-dependent learning ef-fects in which the actual processing time pjr of job Jj when it is scheduled in position r in a processing sequence is defined as

pjr ¼ pjra;

where pj is the normal processing time of job Jj and a (a < 0) is the learning index. He showed that the single-machine sched-uling problems under the proposed model are polynomially solvable if the objective is to minimize the deviation from acommon due date or to minimize the sum of flow times. Mosheiov [15] investigated several other single-machine schedulingproblems, and the problem of minimizing the total flow time on identical parallel machines. Lee et al. [12] considered a bicri-terion single-machine scheduling problem under the above learning effect model. The objective is to find a sequence thatminimizes a linear combination of the total completion time and the maximum tardiness. Liu et al. [13] proved that theweighted shortest processing time (WSPT) rule, the earliest due date (EDD) rule and the modified Moore–Hodgson algorithm

. All rights reserved.

7.n), [email protected] (D. Xu).

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2210 Y. Yin et al. / Information Sciences 181 (2011) 2209–2217

can, under certain conditions, construct the optimal schedule for the problem to minimize the following three objectives: thetotal weighted completion time, the maximum lateness and the number of tardy jobs, respectively. They also gave an errorestimation for each of these rules for the general cases. Wang and Xia [21] considered the flow shop scheduling problemswith a learning effect analogous to Biscup’s model. The objective is to minimize the makespan and total flow time. They gavea heuristic algorithm with a worst-case error bound of m for each criterion, where m is the number of machines. They alsofound polynomial time solutions to two special cases of the problems, i.e., identical processing times on each machine and anincreasing series of dominating machines. Xu et al. [23] further considered the flow shop scheduling problems with theobjective of minimizing the three regular performance criteria, i.e., total weighted completion time, discounted totalweighted completion time, and sum of the quadratic job completion times. They presented approximation algorithms byusing the optimal permutations for the corresponding single-machine scheduling problems and analyzed the worst-case er-ror bounds of the algorithms.

Note that position-dependent learning effects neglect the processing times of the jobs already processed. If human inter-actions have significant impacts during the processing of jobs, the processing times will be added to the employees’ expe-rience and thus cause learning effects. For situations like this it might be more appropriate to consider a time-dependentlearning effect [3]. Kuo and Yang [11] suggested to model learning as follows:

pjr ¼ pj 1þXr�1

k¼1

p½k�

!a

;

where a 6 0 denotes the learning index and p[k] represents the normal processing time of a job when it is scheduled in posi-tion k in the sequence. They showed that the single-machine scheduling problem to minimize the total complete time ispolynomially solvable under the proposed model. Wang et al. [20] further considered several other single-machine sched-uling problem. They showed by several examples that the total weighted completion time minimization problem, the max-imum lateness minimization problem and the number of tardy jobs minimization problem cannot be optimally solved by thecorresponding classical scheduling rules. But for some special cases, the problems can be solved in polynomial time. Theyalso used the classical rules as heuristic algorithms for these three general problems, respectively, and analyzed theirworst-case error bounds. Besides, Koulamas and Kyparisis [10] considered a different time-dependent learning effect asfollows:

pjr ¼ pj 1�Pr�1

k¼1p½k�Pnk¼1pk

!a

;

where a P 1 denotes the learning index. They showed that the single-machine makespan and total completion time mini-mization problems under the proposed model are polynomially solvable. Cheng et al. [7] considered the following time-dependent learning effect model:

pjr ¼ pj 1þXr�1

l¼1

ln p½l�

!a

with the assumption that lnpj P 1 for all job Jj, where a 6 0 is the learning index.Recently, Wu and Lee [22] proposed a new learning effect model where the actual job processing time not only depends

on its scheduled position, but also depends on the sum of the processing times of the jobs already processed. Their model canbe described as follows:

pjr ¼ pj 1þPr�1

k¼1p½k�Pnk¼1pk

!a1

ra2 ;

where a1 and a2 denote two learning indices with a1 < 0 and a2 < 0. They showed that the single-machine makespan and totalcompletion time minimization problems are polynomially solvable and that the total weighted completion time minimiza-tion problem has a polynomial optimal solution under certain agreeable conditions. Cheng et al. [6] considered a similarmodel that can be described as follows:

pjr ¼ pj 1�Pr�1

k¼1p½k�Pnk¼1pk

!a1

ra2 ;

where a1 and a2 denote two learning indices with a1 P 1 and a2 < 0. They obtained similar results as in Wu and Lee [22]. Inaddition, they presented polynomial optimal solutions for some special cases of the m-machine flowshop problems to min-imize the makespan and total completion time. Yin et al. [28] developed a more general model with learning effects asfollows:

pjr ¼ pjfXr�1

k¼1

p½k�

!gðrÞ; r ¼ 1;2; . . . ;n;

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Y. Yin et al. / Information Sciences 181 (2011) 2209–2217 2211

where f : [0,+1) ? (0,1] is a differentiable non-increasing function with f0 is non-decreasing on [0,+1) and f(0) = 1, and g :[1,+1) ? (0,1] is a non-increasing function with g(1) = 1. This model is a significant generalization of some of the learningeffect models in the literature. In particular, it was shown that some single-machine scheduling problems and m-machinepermutation flowshop problems are still polynomially solvable under the proposed model. The idea was further studiedin Yin et al. [29,30].

For recent results and trends in scheduling problems with learning effect, the reader may refer to the recent review paperof Biskup [3]. For more examples we refer the reader a book by Gawiejnowicz [8].

As a continuation of Yin et al. [28], we further investigate some single-machine scheduling problems under the generallearning effect model. The remainder of this paper is organized as follows. In Section 2, we formulate the general model. InSection 3, we consider the solution procedures for the single-machine scheduling problems to minimize the total weightedcompletion time, maximum lateness, total tardiness, total lateness, discounted total weighted completion time, sum of ear-liness penalties and weighted sum of earliness penalty. Some conclusions are given in the last Section.

2. Notations and assumptions

Assume that there are n jobs J1, J2, . . . , Jn to be processed on a single machine. The machine can handle one job at a time andpreemption is not allowed. Each job Jj has a normal processing time pj, a due date dj and a positive weight wj. If job Jj is sched-uled in position r in a processing sequence, then its actual processing time is defined as

pjr ¼ pjfXr�1

k¼1

p½k�

!gðrÞ; r ¼ 1;2; . . . ; n; ð1Þ

whereP0

k¼1p½k� ¼ 0; p½k� denotes the normal processing time of the job scheduled in position k in the sequence, f :[0,+1) ? (0,1] is a differentiable non-increasing function with f0 is non-decreasing on [0,+1) and f(0) = 1, and g :[1,+1) ? (0,1] is a non-increasing function with g(1) = 1. For convenience, we denote the learning effect given in (1) by LEgtp

as in Yin et al. [28]. In such a learning effect model, we can see that the actual processing time of a job depends not only onthe total processing time of the jobs already processed, but also on its scheduled position. It is also evident from the modelthat the longer the already processed jobs or the later the job position is, the stronger the learning effect is on the subsequentjobs that are yet to be processed.

For a given schedule p, let Cj(p), Lj(p) = Cj(p) � dj, Tj(p) = max{Lj(p),0} and Ej(p) = max{dj � Cj(p),0} denote the completetime, lateness, tardiness and earliness of job Jj, respectively. From (1), the completion time of the job scheduled in position rin p, denoted as C[r](p), is

C ½r�ðpÞ ¼Xr

k¼1

p½k�fXk�1

l¼1

p½l�

!gðkÞ: ð2Þ

In the sequel, we will adopt the three-field notation scheme ajbjc introduced by Graham et al. [9] to denote the consid-ered problems in this paper. We consider the following seven regular objectives:

PwjCj (total weighted completion time),

Lmax (maximum lateness),P

Lj (total lateness),P

Tj (total tardiness),P

wjð1� e�rCj Þ (discounted total weighted completiontime),

PhðEjÞ (sum of earliness penalties) and

PwjhðEjÞ (total weighted earliness penalty), where r 2 (0,1) is the discount

factor and h is a strictly increasing function.

3. Some single-machine scheduling problems

3.1. The total weighted completion time and maximum lateness minimization problems

For the single-machine makespan and sum of the kth power of completion times minimization problems under model (1),Yin et al. [28] have shown that an optimal schedule can be obtained by the shortest processing time (SPT) rule. For the single-machine total weighted completion time and maximum lateness minimization problems, an optimal schedule can be

obtained by the WSPT rule i:e:; sequencing the jobs in non-decreasing order of pj

wj

� �and the EDD rule, respectively [18].

However, Mosheiov [15] showed that the polynomial optimal solutions of the classical version do not hold when learningis considered. Hence they do not hold under model (1) too.

Now we turn our attention to obtain schedules whose performance approximates that of optimal schedules. In order tosolve the problem approximately, we can use the WSPT rule as a heuristic algorithm for the general problem 1jLEgtpj

PwjCj.

The performance of the algorithm is evaluated by its worst-case bound.

Theorem 3.1. Let S⁄ be an optimal schedule and S be the WSPT sequence for the problem 1jLEgtpjP

wjCj. Then

q1 ¼P

wjCjðSÞPwjCjðS�Þ

61

f ðP

pj � pminÞgðnÞ;

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2212 Y. Yin et al. / Information Sciences 181 (2011) 2209–2217

where pmin = min{pjjj = 1,2, . . . ,n}.

Proof. Without loss of generality, suppose that p1w16

p2w26 � � � 6 pn

wn. Since 0 < f(x) 6 1 and 0 < g(y) 6 1 for all x 2 [0,1) and

y 2 [1,1), respectively, we have

XwjCjðSÞ ¼ w1p1 þw2ðp1 þ p2f ðp1Þgð2ÞÞ þ � � � þwn

Xn

k¼1

pkfXk�1

l¼1

pl

!gðkÞ 6 w1p1 þw2ðp1 þ p2Þ þ � � � þwn

Xn

k¼1

pk

¼Xn

j¼1

wj

Xj

k¼1

pk

!;

wherePn

j¼1wjPj

k¼1pk

� �is the optimal objective value of the classical version of the problem.

Let (J[1], J[2], . . . , J[n]) be the order in which jobs complete in the optimal schedule S⁄. Since both f and g are non-increasing,we have

XwjCjðS�Þ ¼ w½1�p½1� þw2ðp½1� þ p2f ðp½1�Þgð2ÞÞ þ � � � þwn

Xn

k¼1

p½k�fXk�1

l¼1

p½l�

!gðkÞ ¼

Xn

j¼1

w½j�Xj

k¼1

p½k�fXk�1

l¼1

p½l�

!gðkÞ

!

PXn

j¼1

w½j�Xj

k¼1

p½k�

!fXn�1

l¼1

p½l�

!gðnÞP

Xn

j¼1

w½j�Xj

k¼1

p½k�

!fXn

l¼1

p½l� � pmin

!gðnÞ

PXn

j¼1

wj

Xj

k¼1

pk

!fXn

l¼1

pl � pmin

!gðnÞ:

Hence,

q1 ¼P

wjCjðSÞPwjCjðS�Þ

61

fP

pj � pmin

� �gðnÞ

:

It is not difficult to see that if f(x)=1 and g(y) = 1 for all x 2 [0,1) and y 2 [1,1), we haveP

wjCjðSÞPwjCjðS�Þ

¼ 1: This result is intuitive

because, when f(x) = 1 and g(y) = 1, the WSPT sequence is optimal. h

Although the WSPT sequence does not provide the optimal schedule for 1jLEgtpjP

wjCj, it is still optimal if jobs have re-versely agreeable weights, i.e., pi 6 pj implies wi P wj for all jobs Ji and Jj as shown in Yin et al. [28].

Theorem 3.2 [28]. For the problem 1jLEgtpjP

wjCj, if jobs have reversely agreeable weights, i.e., pi 6 pj implies wi P wj for alljobs Ji and Jj, then there exists an optimal schedule in which the jobs are ordered according to the WSPT rule.

Next we use the EDD rule as a heuristic algorithm for the problem 1jLEgtpjLmax. To develop a worst-case performance ratio

for the algorithm, we have to avoid problems caused by non-positive Lmax. As suggested by Kise et al. [14] and Cheng andWang [5], this can be achieved by simply adding a constant value at least as large as the maximum due date, denoted as

dmax, to Lmax. Hence the worst-case bound can be determined by the ratio LmaxðSÞþdmaxLmaxðS�Þþdmax

, where S and Lmax(S) denote the schedule

and the corresponding maximum lateness, respectively, while S⁄ and Lmax(S⁄) denote the optimal schedule and the minimummaximum lateness value, respectively, and dmax = max{djjj = 1,2, . . . ,n}.

Theorem 3.3. Let S⁄ be an optimal schedule and S be the EDD sequence for the problem 1jLEgtpjLmax. Then q2 ¼LmaxðSÞþdmaxLmaxðS�Þþdmax

6

Ppj

C�max,

where C�max is the optimal makespan of the problem 1jLEgtpjCmax.

Proof. Without loss of generality, suppose that d1 6 d26� � �6 dn. Since 0 < f(x) 6 1 and 0 < g(y) 6 1 for all x 2 [0,1) andy 2 [1,1), respectively, we have

LmaxðSÞ ¼maxXj

k¼1

pkfXk�1

l¼1

pl

!gðkÞ � djjj ¼ 1;2; . . . ;n

( )6 max

Xj

k¼1

pk � djjj ¼ 1;2; . . . ;n

( );

where maxPj

k¼1pk � djjj ¼ 1;2; . . . ;nn o

is the optimal objective value of the classical version of the problem.

Let (J[1], J[2], . . . , J[n]) be the order in which jobs complete in the optimal schedule S⁄. We have

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Y. Yin et al. / Information Sciences 181 (2011) 2209–2217 2213

LmaxðS�Þ ¼maxXj

k¼1

p½k�fXk�1

l¼1

p½l�

!gðkÞ � d½j�jj ¼ 1;2; . . . ;n

( )

¼maxXj

k¼1

p½k� � d½j� �Xj

k¼1

p½k� þXj

k¼1

p½k�fXk�1

l¼1

p½l�

!gðkÞjj ¼ 1;2; . . . ;n

( )

P maxXj

k¼1

p½k� � d½j�jj ¼ 1;2; . . . ;n

( )�Xn

k¼1

p½k� þXn

k¼1

p½k�fXk�1

l¼1

p½l�

!gðkÞ

P maxXj

k¼1

p½k� � d½j�jj ¼ 1;2; . . . ;n

( )�Xn

k¼1

pk þ C�max;

where C�max can be obtained by the SPT sequence (see [[28], Theorem 2.5]), hence LmaxðSÞ � LmaxðS�Þ 6Pn

k¼1pk � C�max and so

q2 ¼LmaxðSÞ þ dmax

LmaxðS�Þ þ dmax6

LmaxðS�Þ þPnk¼1

pk � C�max þ dmax

LmaxðS�Þ þ dmax¼ 1þ

Pnk¼1pk � C�max

LmaxðS�Þ þ dmax6 1þ

Pnk¼1pk � C�max

C�max¼P

pj

C�max:

It is not difficult to see that if f(x) = 1 and g(y) = 1 for all x 2 [0,1) and y 2 [1,1), we have C�max ¼P

pj and so q2 = 1. Thisresult is intuitive because, when f(x) = 1 and g(y) = 1, the EDD sequence is optimal. h

Although the EDD sequence does not provide an optimal schedule for 1jLEgtpjLmax, it is still optimal if the job processingtimes and due dates are agreeable, i.e., di 6 dj implies pi 6 pj for all jobs Ji and Jj.

Theorem 3.4 [28]. For the problem 1jLEgtpjLmax, if the job processing times and due dates are agreeable, i.e., di 6 dj implies pi 6 pj

for all jobs Ji and Jj, then there exists an optimal schedule in which the jobs are ordered according to the EDD rule.

3.2. The total lateness and total tardiness minimization problems

In this section, we study the single-machine total lateness and total tardiness minimization problems under model (1).For 1jLEDgtpj

PLj, we have the following main result.

Theorem 3.5. For the problem 1jLEgtpjP

Lj, there exists an optimal schedule in which the jobs are ordered according to the SPTrule.

Proof. For any given feasible schedule, the total latenessP

Lj ¼Pn

r¼1ðC½r� � d½r�Þ ¼Pn

r¼1C½r� �Pn

r¼1d½r�. SincePn

r¼1d½r� is a con-stant,

PLj is minimized if

Pnr¼1C½r� ¼

PCj is minimized. By Corollary 2.7 in [28],

PCj is minimized by the SPT sequence and

so the SPT sequence leads to an optimal schedule for 1jLEgtpjP

Lj. h

It is well known that the single-machine total tardiness problem is NP-hard. Hence 1jLEgtpjP

Tj must be NP-hard, too. Butit is still optimal if the job processing times and the due dates are agreeable.

Theorem 3.6. For the problem 1jLEgtpjP

Tj, if the job processing times and due dates are agreeable, i.e., di 6 dj implies pi 6 pj forall jobs Ji and Jj, then there exists an optimal schedule in which the jobs are ordered according to the EDD rule.

Proof. By pairwise job interchange argument. Suppose that there exists an optimal schedule S = (p1JjJip2) where p1 and p2

denote the partial sequences of S. Let S0 be a schedule with jobs Ji and Jj of S mutually exchanged, i.e., S0 = (p1JiJjp2). In addi-tion, we assume that di 6 dj. Since the job processing times and due dates are agreeable, we have pi 6 pj. We will show thatthe interchange of jobs Ji and Jj does not increase the objective value. The repeated implementation of this argument will leadto the optimality of the EDD rule for the problem 1jLEgtpj

PTj. Specifically, it suffices to show that Ti(S0) + Tj(S0) 6 Tj(S) + Ti(S),

i.e., max{Li(S0),0} + max{Lj(S0),0} 6max{Lj(S),0} + max{Li(S),0}. By the proof of Theorems 2.5, 2.6 and 2.11 in [28], we haveCi(S0) 6 Cj(S0) 6 Ci(S), Ci(S0) 6 Cj(S), Lj(S0) 6 Li(S) and Li(S0) 6 Li(S), and so Ci(S0) + Cj(S0) 6 Cj(S) + Ci(S). Now, we consider the fol-lowing cases.

Case 1: Li(S) 6 0. Then Li(S0) 6 Li(S) 6 0 and Lj(S0) 6 Li(S) 6 0. Hence Tj(S) + Ti(S) P 0 = Ti(S0) + Tj(S0).Case 2: Li(S) > 0 and Lj(S) 6 0. Then Tj(S) + Ti(S) = Li(S). Now we have the following cases:

Case 2.1: Li(S0) P 0 and Lj(S0) P 0. Since Lj(S) 6 0, i.e., Cj(S) 6 dj, we have

TjðSÞ þ TiðSÞ � TiðS0Þ � TjðS0Þ ¼ LiðSÞ � LiðS0Þ � LjðS0Þ ¼ CiðSÞ � di � ðCiðS0Þ � diÞ � ðCjðS0Þ � djÞ¼ dj � ðCiðS0Þ þ CjðS0Þ � CiðSÞÞP dj � CjðSÞP 0:

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2214 Y. Yin et al. / Information Sciences 181 (2011) 2209–2217

Case 2.2: Li(S0) 6 0 and Lj(S0) P 0. Then

TjðSÞ þ TiðSÞ � TiðS0Þ � TjðS0Þ ¼ LiðSÞ � LjðS0Þ ¼ CiðSÞ � di � ðCjðS0Þ � djÞ ¼ ðCiðSÞ � CjðS0ÞÞ þ ðdj � diÞP 0:

Case 2.3: Li(S0) P 0 and Lj(S0) 6 0. Then

TjðSÞ þ TiðSÞ � TiðS0Þ � TjðS0Þ ¼ LiðSÞ � LiðS0Þ ¼ CiðSÞ � di � ðCiðS0Þ � diÞ ¼ CiðSÞ � CiðS0ÞP 0:

Case 2.4: Li(S0) 6 0 and Lj(S0) 6 0. Then

TjðSÞ þ TiðSÞ � TiðS0Þ � TjðS0Þ ¼ LiðSÞ > 0:

Case 3: Li(S) > 0 and Lj(S) > 0. Analogous to the proof of Case 2, we have Tj(S) + Ti(S) � Ti(S0) � Tj(S0) P 0.

Thus, in any case, we have Tj(S) + Ti(S) P Tj(S0) + Ti(S0). h

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

Corollary 3.7. For the problem 1jLEgtp; pj ¼ pjP

Tj, there exists an optimal schedule in which the jobs are ordered according to theEDD rule.

Let k be a positive real number. If dj = kpj for all 1 6 j 6 n, then the job processing times and due dates are agreeable.Hence, we have the following corollary.

Corollary 3.8. For the problem 1jLEgtp; dj ¼ kpjjP

Tj, where k is a positive real number, there exists an optimal schedule in whichthe jobs are ordered according to the EDD rule.

3.3. The discounted total weighted completion time minimization problem

Pinedo [17] considered the single-machine discounted total weighted completion time minimization problem, denoted as1kP

wjð1� e�rCj Þ, and showed that for this problem, an optimal schedule can be obtained by the weighted discounted short-

est processing time (WDSPT) rule, i.e., sequencing the jobs in non-decreasing order of 1�e�rpj

wje�rpj . However, the WDSPT sequence

does not yield an optimal schedule for the general problem 1jLEgtpjP

wjð1� e�rCj Þ, as shown in the following example.

Example 3.9. Let f(x) = (1 + x)�2 and g(x) = x�2. Given n = 2, p1 = 2, p2 = 4, w1 = 1, w2 = 4 and r = 0.5. The WDSPT sequence (J2,J1) yields a value of 3.9452, while the sequence (J1, J2) yields the optimal objective value of 2.7799.

In order to solve the problem approximately, we can use the WDSPT rule as a heuristic algorithm for the general problem1jLEgtpj

Pwjð1� e�rCj Þ. The performance of the algorithm is evaluated by its worst-case bound. Before proceeding, we first

formulate a useful lemma.

Lemma 3.10. 1 � e�aa P a(1 � e�a) for all 0 6 a 6 1 and a 2 (�1, +1).

Proof. Let F(a) = (1 � e�aa) � a(1 � e�a). Taking the first and second derivatives of F(a) with respect to a, respectively, wehave

F 0ðaÞ ¼ ae�aa � ð1� e�aÞ

and

F 00ðaÞ ¼ �a2e�aa6 0:

Hence F(a) is a concave function on 0 6 a 6 1. In addition, F(0) = F(1) = 0, hence F(a) P 0. h

Theorem 3.11. Let S⁄ be an optimal schedule and S be the WDSPT sequence for the problem 1jLEgtpjP

wjð1� e�rCj Þ. Then

q3 ¼P

wj 1� e�rCjðSÞ� �

Pwj 1� e�rCjðS�Þ� � 6 1

fP

pj � pmin

� �gðnÞ

;

where pmin = min{pjjj = 1,2, . . . ,n}.

Proof. Without loss of generality, we can suppose that 1�e�rp1

w1e�rp1 61�e�rp2

w2e�rp2 6 � � � 6 1�e�rpn

wne�rpn . Since 0 < r < 1, 0 < f(x) 6 1 and0 < g(y) 6 1 for all x 2 [0,1) and y 2 [1,1), we have

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Y. Yin et al. / Information Sciences 181 (2011) 2209–2217 2215

Xwj 1� e�rCjðSÞ� �

¼ w1ð1� e�rp1 Þ þw2 1� e�rðp1þp2 f ðp1Þgð2ÞÞ� �

þ � � � þwn 1� e�rPn

k¼1

pkfPk�1

l¼1

pl

� �gðkÞ

� �0B@

1CA

6 w1ð1� e�rp1 Þ þw2 1� e�rðp1þp2Þ� �

þ � � � þwn 1� e�rPn

k¼1

pk

0@

1A ¼Xn

j¼1

wj 1� e�rPj

k¼1

pk

0B@

1CA;

wherePn

j¼1wj 1� e�rPj

k¼1pk

� �is the optimal objective value of the classical version of the problem.

Let (J[1], J[2], . . . , J[n]) be the order in which jobs complete in the optimal schedule S⁄. Since both f and g are non-increasing,by Lemma 3.10, we have

Xwj 1� e�rCjðS�Þ� �

¼ w½1� 1� e�rp½1�ð Þ þw½2� 1� e�rðp½1�þp½2� f ðp½1� Þgð2ÞÞ� �

þ � � � þw½n� 1� e�rPn

k¼1

p½k�fPk�1

l¼1

p½l�

� �gðkÞ

� �0B@

1CA

¼Xn

j¼1

w½j� 1� e�rPj

k¼1

p½k� fPk�1

l¼1

p½l�

� �gðkÞ

� �0B@

1CA P

Xn

j¼1

w½j� 1� e�rPj

k¼1

p½k�

� �fPn�1

l¼1

p½l�

� �gðnÞ

0B@

1CA

PXn

j¼1

w½j�fXn�1

l¼1

p½l�

!gðnÞ 1� e

�rPj

k¼1

p½k�

0B@

1CA since 0 < f

Xn�1

l¼1

p½l�

!gðnÞ 6 1

!

¼ fXn�1

l¼1

p½l�

!gðnÞ

Xn

j¼1

w½j� 1� e�rPj

k¼1

p½k�

0B@

1CAP f

Xn

l¼1

pl � pmin

!gðnÞ

Xn

j¼1

wj 1� e�rPj

k¼1

pk

0B@

1CA:

Hence,

q3 ¼P

wjð1� e�rCjðSÞÞPwjð1� e�rCjðS�ÞÞ

61

fP

pj � pmin

� �gðnÞ

:

It is not difficult to see that if f(x)=1 and g(y) = 1 for all x 2 [0,1) and y 2 [1,1), we have q3 ¼P

wjð1�e�rCjðSÞÞPwjð1�e

�rCj ðS�ÞÞ¼ 1. This result is

intuitive since, when f(x) = 1 and g(y) = 1, the WDSPT sequence is optimal. h

Although the WDSPT sequence does not provide the optimal schedule for 1jLEgtpjP

wj 1� e�rCj� �

, it is still optimal if jobshave reversely agreeable weights.

Theorem 3.12. For the problem 1jLEgtpjP

wjð1� e�rCj Þ, if jobs have reversely agreeable weights, i.e., pi 6 pj implies wi P wj for alljobs Ji and Jj, then there exists an optimal schedule in which the jobs are ordered according to the WDSPT rule.

Proof. We still adopt the same notations as in the proof of Theorem 3.6. Suppose that there is an optimal schedule whichdoes not follow the WDSPT rule, i.e., there exist at least two adjacent jobs, say job Ji and job Jj, such that Jj is scheduled before

Ji with 1�e�rpj

wje�rpj >

1�e�rpi

wie�rpi . Note that the weights of the two jobs are reversely agreeable by assumption. Thus, we have pj P pi and

wj 6 wi. Hence Ci(S0) 6 Cj(S) and Cj(S0) 6 Ci(S) by the proof of Theorem 2.5 in [28]. Next we will show that the interchange ofjobs Ji and Jj does not increase the objective value. The repeated implementation of this argument will lead to the optimalityof the WDSPT rule for the problem 1jLEgtpj

Pwjð1� e�rCj Þ. Specially, it suffices to show that

wið1� e�rCiðS0 ÞÞ þwjð1� e�rCjðS0ÞÞ 6 wið1� e�rCiðSÞÞ þwjð1� e�rCjðSÞÞ:

In fact, since r 2 (0,1), Ci(S0) 6 Cj(S) and Cj(S0) 6 Ci(S), we have

wið1� e�rCiðSÞÞ þwjð1� e�rCjðSÞÞ �wið1� e�rCiðS0 ÞÞ �wjð1� e�rCjðS0ÞÞ ¼ wie�rCiðS0Þ þwje�rCjðS0 Þ �wie�rCiðSÞ �wje�rCjðSÞ

P wie�rCjðSÞ þwje�rCiðSÞ �wie�rCiðSÞ �wje�rCjðSÞ ¼ ðwi �wjÞ e�rCjðSÞ � e�rCiðSÞ� �

P 0: �

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

Corollary 3.13. For the problem 1jLEgtp; pj ¼ pjP

wjð1� e�rCj Þ, there exists an optimal schedule in which the jobs are ordered innon-decreasing order of their weights.

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2216 Y. Yin et al. / Information Sciences 181 (2011) 2209–2217

3.4. Common due date problems

In this section we consider the problems to minimize the total earliness penalty and total weighted earliness penalty. Forthe classical scheduling problems with common due date, there are some results in [4]. We assume that all jobs have a com-mon due date d and a common release time 0. For a given schedule p, the sum of earliness penalties is EðpÞ ¼

Pnj¼1hðEjÞ,

where h is a strictly increasing function and Ej = max{d � Cj,0} is the earliness of job Jj in p. The problems are to find optimalschedules to minimize 1jLEgtpj

PhðEjÞ and 1jLEgtpj

PwjhðEjÞ. Since the problems are to minimize earliness penalties, the jobs

should be processed as late as possible. It is obvious that in the optimal schedule the completion time of the last job is d ifthere are no tardy jobs.

Theorem 3.14. For the problem 1jLEgtpjP

hðEjÞ, there exists an optimal schedule in which the jobs are ordered according to thelongest processing time (LPT) rule, where the starting time of the first job is

t0 ¼ d�Xn

k¼1

p½k�fXk�1

l¼1

p½l�

!gðkÞ

if no tardy jobs and the schedule (J[1], J[2], . . . , J[n]) is sequenced according to the LPT rule.

Proof. Suppose that there exists an optimal schedule S = (p1JjJi p2) with pj < pi, where p1 and p2 denote the partial sequencesof S. Let S0 be a schedule with jobs Ji and Jj of S mutually exchanged, i.e., S0 = (p1JiJjp2). We will show that the interchange ofjobs Ji and Jj does not increase the objective value. The repeated implementation of this argument will lead to the optimalityof the LPT rule for the problem 1jLEgtpj

PhðEjÞ. Specifically, it suffices to show that Ej(S0) = max{d � Cj(S0),0} 6 Ei(S) =

max{d � Ci(S),0} since h is a strictly increasing function. And this can be easily obtained in a similar way as the proof ofTheorem 2.5 in [28]. h

Theorem 3.15. For the problem 1jLEgtpjP

wjhðEjÞ, if jobs have reversely agreeable weights, then there exists an optimal schedulein which the jobs are ordered in non-increasing order of pj

wj(weighted longest processing time (WLPT) rule) and the starting time of

the first job is

t0 ¼ d�Xn

k¼1

p½k�fXk�1

l¼1

p½l�

!gðkÞ

if no tardy jobs and the schedule (J[1], J[2], . . . , J[n]) is sequenced according to the WLPT rule.

Proof. We still adopt the same notations as in the proof of Theorem 3.14. Suppose that there is an optimal schedule whichdoes not follow the WLPT rule, i.e., there exist at least two adjacent jobs, say job Ji and job Jj, such that Jj is scheduled before Ji

with pj

wj< pi

wi. Note that the weights of the two jobs are reversely agreeable by assumption. Thus, we have pj 6 pi and wj P wi.

Hence Ej(S0) 6 Ei(S) by Theorem 3.14. Next we will show that the interchange of jobs Ji and Jj does not increase the objectivevalue. The repeated implementation of this argument will lead to the optimality of the WLPT rule for the problem1jLEgtpj

PwjhðEjÞ. Specially, it suffices to show that wih(Ei(S0)) + wjh(Ej(S0)) 6 wih(Ei(S)) + wjh(Ej(S)). In fact, let B ¼

Pr�1k¼1p½k�

and let B0 denote the completion time of the last job in p1. It follows from pj 6 pi that Ei(S0) = max{d � Ci(S0),0} 6max{d �Cj(S),0} = Ej(S). Now since h is a strictly increasing function, we have

wihðEiðSÞÞ þwjhðEjðSÞÞ �wihðEiðS0ÞÞ �wjhðEjðS0ÞÞP wihðEjðS0ÞÞ þwjhðEiðS0ÞÞ �wihðEiðS0ÞÞ �wjhðEjðS0ÞÞ¼ ðwi �wjÞðhðEjðS0ÞÞ � hðEiðS0ÞÞÞP 0;

as required. h

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

Corollary 3.16. For the problem 1jLEgtp; pj ¼ pjP

wjhðEjÞ, there exists an optimal schedule in which the jobs are ordered in non-increasing order of their weights.

4. Conclusions

Scheduling problems with learning effects have attracted growing attention of the scheduling research community. Yinet al. [28] developed a general model with learning effects which is a significant generalization of some of the learning effectmodels in the literature. This paper provided a continuation of the idea presented by Yin et al. [28]. It further investigatedsome single-machine scheduling problems under the general learning effect model. For each of the following three objec-

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Y. Yin et al. / Information Sciences 181 (2011) 2209–2217 2217

tives, total weighted completion time, maximum lateness and discounted total weighted completion time, it presented anapproximation algorithm which is based on the optimal algorithm for the corresponding single-machine scheduling problemand analyzes its worst-case bound. It showed that the single-machine scheduling problems remain polynomially solvable ifthe objectives are to minimize the total lateness and the sum of earliness penalties. It also showed that under certain con-ditions, the problems to minimize the total tardiness, discounted total weighted completion time and total weighted earli-ness penalties have polynomial time solutions. The results obtained are significant compared with the prior work of using alearning effect model in single-machine scheduling. It is useful to guide practitioners to choose the right scheduling rules andsuitable learning effect model in practice.

Acknowledgements

The authors would like to express our warmest thanks to the referees for their interest in our work and their valuablecomments for improving the paper.

This research was supported in part by the Natural Science Foundation of Jiangxi, China (2010GQS0003); the ScienceFoundation of Education Committee of Jiangxi for Young Scholars, China (GJJ11143, GJJ11144) and the Doctor Fund of EastChina Institute of Technology.

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