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Some scheduling problems with general position-dependent and time-dependent learning effects
Information Sciences 179 (2009) 2416–2425
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Some scheduling problems with general position-dependent and time-dependent learning effects Yunqiang Yin a, Dehua Xu a, Kaibiao Sun b, Hongxing Li b,* a b
College of Mathematics and Information Sciences, East China Institute of Technology, Fuzhou, Jiangxi 344000, China School of Electronic and Information Engineering, Dalian University of Technology, Dalian, Liaoning 116024, China
a r t i c l e
i n f o
Article history: Received 21 April 2008 Received in revised form 11 January 2009 Accepted 26 February 2009
Keywords: Scheduling Learning effect Time-dependent Position-dependent Single-machine Flowshop
a b s t r a c t In scheduling problems with learning effects, most of the research is based on specific learning functions. In this paper, we develop a general model with learning effects where the actual processing time of a job is not only a function of the total normal processing times of the jobs already processed, but also a function of the job’s scheduled position. In particular, it is shown that some single machine scheduling problems and m-machine permutation flowshop problems are still polynomially solvable under the proposed model. These results are significant extensions of some of the existing results on learning effects in the literature. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Scheduling problems have received considerable attention since the middle of the last century. However, most researchers assume that job processing times are known and fixed during the whole processing process (see, e.g., [23,25,35–38]). 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 (see, e.g., [2,3]). During the last few years, learning effect has attracted growing attention in the scheduling community on account of its significance. There have been many attempts to formulate learning effect in a quantitative form as a function of learning variables, called a learning curve. Most of the concepts assume that the learning curve is a non-increasing function which depends on the jobs already performed. For a survey on learning curves, the reader is refereed to Jaber and Bonney [12]. To the best of our knowledge, Biskup [3] and Cheng and Wang [8] are among the pioneers that brought the concept of learning effect into the field of scheduling. Since then, scheduling problems with learning effects have received considerable attention. According to a recent state-of-the-art literature review paper of Biskup [4], over ten learning effect models have been proposed and investigated since 1999. One can distinguish these models into three main categories, that is, positiondependent (see, e.g., [1,3,5,8,22,30]), time-dependent (see, e.g., [13,14]) and a combination of the two (see, e.g., [9,32]). In what follow, we will briefly review some of these models. In the classical position-dependent learning effect model (Biskup [3]), the actual processing time pir of job J i when it is scheduled in the rth position in a processing sequence is defined as * Corresponding author. Tel.: +86 411 84706402; fax: +86 411 84706405. E-mail addresses: [email protected] (Y. Yin), [email protected] (D. Xu), [email protected] (K. Sun), [email protected] (H. Li). 0020-0255/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2009.02.015
Y. Yin et al. / Information Sciences 179 (2009) 2416–2425
pir ¼ pi r a ;
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ð1Þ
where a is a non-positive learning index and pi denotes the normal processing time of job J i . Biskup [3] showed that single machine scheduling problems under model (1) are polynomially solvable if the objective is to minimize the deviation from a common due date or to minimize the sum of flow times. Many researchers have studied such a learning effect model and its variants thereafter, a sample of these papers include [1,6–8,10,16–21,27,28,30,31,33,34,39]. Note that position-dependent learning effects neglect the processing times of the jobs already processed. If human interactions have significant impacts during the processing of jobs, the processing times will be added to the employees’ experience and thus cause learning effects. For situations like this it might be more appropriate to consider a time-dependent learning effect [4]. Kuo and Yang [14] proposed the following time-dependent learning effect model:
pir ¼ pi 1 þ
!a
r1 X
p½k
ð2Þ
;
k¼1
where a 6 0 denotes the learning index and p½k represents the normal processing time of a job when it is scheduled in the kth position in the sequence. Kuo and Yang [14] showed that the single machine scheduling problem to minimize total complete time is polynomially solvable under model (2). For more papers about this time-dependent learning effect model, the reader is refereed to [15,16,29]. Besides, Koulamas and Kyparisis [13] considered the following time-dependent learning effect model:
Pr1
k¼1 p½k
!a
pir ¼ pi 1 Pn
ð3Þ
;
k¼1 pk
where a P 1 denotes the learning index. They showed that the single machine makespan and total completion time minimization problems under model (3) are polynomially solvable. Recently, Wu and Lee [32] 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 can be described as follows:
Pr1
k¼1 p½k Pn k¼1 pk
pir ¼ pi 1 þ
!a 1 r a2 ;
ð4Þ
where a1 < 0 and a2 < 0 denote two learning indices. They showed that the single machine makespan and the total completion time problems are polynomially solvable under model (4) and that the total weighted completion time has a polynomial optimal solution under certain agreeable conditions. Cheng et al. [9] consider a similar model that can be described as follows:
Pr1 pir ¼ pi 1
p½k Pk¼1 n k¼1 pk
!a 1 r a2 ;
ð5Þ
where a1 and a2 denote two learning indices with a1 P 1 and a2 < 0. They obtained similar results as in Wu and Lee [32]. In addition, they presented polynomial optimal solutions for some special cases of the m-machine flowshop problems to minimize makespan and total completion time. Note that all of the papers mentioned above concerning leaning effects on scheduling are based on specific a problems P a and pir ¼ pi 1 Px 1 r a2 , where x ¼ r1 learning functions, such as pir ¼ pi r a , pir ¼ pi ð1 þ xÞa , pir ¼ pi 1 Px i¼1 p½i and Pn P ¼ i¼1 pi . It is natural to ask whether the above results are still true for a general learning effect model of the form pir ¼ pi f ðxÞgðrÞ, where f and g are two real functions. This paper attempts to answer this question. We show that some single machine scheduling problems and m-machine permutation flowshop problems are still polynomially solvable under the above learning effect model if f : ½0; þ1Þ ! ð0; 1 is a differentiable non-increasing function with f 0 is non-decreasing on ½0; þ1Þ and f ð0Þ ¼ 1, and g : ½1; þ1Þ ! ð0; 1 is a non-increasing function with gð1Þ ¼ 1. These results are significant extensions of some of the existing results on learning effects in the literature. 2. Some single machine problems Assume that there are n jobs J 1 ; J 2 ; . . . ; J n to be processed on a single machine. The machine can handle one job at a time and preemption is not allowed. Each job J j has a normal processing time pj , a due date dj and a positive weight wj . If job J j is scheduled in the rth position in a processing sequence, then its actual processing time is defined as
pjr ¼ pj f
r1 X
! p½k gðrÞ;
r ¼ 1; 2; . . . ; n;
ð6Þ
k¼1
P0 where k¼1 p½k ¼ 0, p½k denotes the normal processing time of the job scheduled in the kth position in the sequence, f : ½0; þ1Þ ! ð0; 1 is a differentiable non-increasing function with f 0 is non-decreasing on ½0; þ1Þ and f ð0Þ ¼ 1, and
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g : ½1; þ1Þ ! ð0; 1 is a non-increasing function with gð1Þ ¼ 1. For convenience, we denote the learning effect given in Eq. (6) by LEgtp . In such a learning effect model, we can see that the actual processing time of a job depends not only on the total processing time of the jobs already processed, but also on its scheduled position. It is also evident from the model that the longer the already processed jobs or the later the job position is, the stronger the learning effect is on the subsequent jobs that are yet to be processed. Remark 2.1. One may note that model (6) is a generalization of models 1–5. Indeed: (i) (ii) (iii) (iv)
Define f : ½0; þ1Þ ! ð0; 1 by f ðxÞ ¼ 1. Then model (6) is a generalization of model (1). Define g : ½1; of models (2) and (3). þ1Þ a ! ð0; 1 by gðxÞ ¼ 1. Then model (6) is a generalization P Let f ðxÞ ¼ 1 þ Px 1 and gðxÞ ¼ xa2 , where a1 < 0, a2 < 0 and P ¼ nk¼1 pk . Then model (6) reduces to model (4). P a Let f ðxÞ ¼ 1 Px 1 and gðxÞ ¼ xa2 , where a1 P 1, a2 < 0 and P ¼ nk¼1 pk . Then model (6) reduces to model (5).
In this section we will study some single machine problems under model (6). Before developing the results, we first give some lemmas which are useful for the following theorems. Lemma 2.2. If f : ½0; þ1Þ ! ð0; 1 is a differentiable non-increasing function and its derivative f 0 is non-decreasing on ½0; þ1Þ, then
ð1 aÞf ðaÞ þ akf ða þ tÞ kf ða þ atÞ 6 0 for a P 1; 0 < k 6 1; a P 0 and t P 0. Proof. Let
FðtÞ ¼ ð1 aÞf ðaÞ þ akf ða þ tÞ kf ða þ atÞ: Taking the first derivative of FðtÞ with respect to t, we have
F 0 ðtÞ ¼ akf 0 ða þ tÞ akf 0 ða þ atÞ ¼ akðf 0 ða þ tÞ f 0 ða þ atÞÞ: Since a P 1, t P 0 and f 0 is non-decreasing, we have f 0 ða þ tÞ f 0 ða þ atÞ 6 0 and so F 0 ðtÞ 6 0. This implies that FðtÞ is nonincreasing on t P 0. Since a P 1; 0 < k 6 1 and f ðaÞ P 0, we have
FðtÞ 6 Fð0Þ ¼ ð1 aÞf ðaÞ þ akf ðaÞ kf ðaÞ ¼ ð1 aÞð1 kÞf ðaÞ 6 0: This completes the proof. h Lemma 2.3. If f : ½0; þ1Þ ! ð0; 1 is a differentiable non-increasing function and its derivative f 0 is non-decreasing on ½0; þ1Þ, then 0
f ðaÞ þ kk2 tf ða þ tÞ kk1 f ða þ tÞ P 0 for 0 < k 6 1, 0 6 k1 6 k2 6 1; a P 0 and t P 0. Proof. Let 0
Fðk1 ; k2 ; tÞ ¼ f ðaÞ þ kk2 tf ða þ tÞ kk1 f ða þ tÞ: Then
oF ok1
¼ kf ða þ tÞ 6 0,
oF ok2
0
¼ ktf ða þ tÞ 6 0. Since 0 6 k1 6 k2 6 1, we have 0
Fðk1 ; k2 ; tÞ P Fð1; 1; tÞ ¼ f ðaÞ þ ktf ða þ tÞ kf ða þ tÞ: By Lagrange mean value theorem, there exists a value n between a and a þ t such that
f ða þ tÞ f ðaÞ ¼ f 0 ðnÞt: Since 0 < k 6 1; t P 0 and f 0 is non-decreasing, we have 0
0
Fðk1 ; k2 ; tÞ P f ðaÞ þ ktf ða þ tÞ kf ða þ tÞ P kf ðaÞ þ ktf ða þ tÞ kf ða þ tÞ ¼ ktðf 0 ða þ tÞ f 0 ðnÞÞ P 0: This completes the proof. h Lemma 2.4. If f : ½0; þ1Þ ! ð0; 1 is a differentiable non-increasing function and its derivative f 0 is non-decreasing on ½0; þ1Þ, then
ða 1Þf ðaÞ þ kk2 f ða þ atÞ akk1 f ða þ tÞ P 0 for a P 1; 0 < k 6 1, 0 6 k1 6 k2 6 1; a P 0 and t P 0.
Y. Yin et al. / Information Sciences 179 (2009) 2416–2425
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Proof. Let
FðaÞ ¼ ða 1Þf ðaÞ þ kk2 f ða þ atÞ akk1 f ða þ tÞ: Taking the first derivative of FðaÞ with respect to a, we have 0
F 0 ðaÞ ¼ f ðaÞ þ kk2 tf ða þ atÞ kk1 f ða þ tÞ: Since a P 1, t P 0 and f 0 is non-decreasing, by Lemma 2.3, we have 0
F 0 ðaÞ P F 0 ð1Þ ¼ f ðaÞ þ kk2 tf ða þ tÞ kk1 f ða þ tÞ P 0: This implies that FðaÞ is non-decreasing on a P 1 and so
FðaÞ P Fð1Þ ¼ kk2 f ða þ tÞ kk1 f ða þ tÞ ¼ kf ða þ tÞðk2 k1 Þ P 0: This completes the proof. h In the sequel, we will adopt the three-field notation scheme ajbjc introduced by Graham et al. [11] to denote the considered problems in this paper. First, we study the single machine makespan minimization problem under model (6), denoted by 1jLEgtp jC max . And we show that an optimal schedule for the problem can be obtained by the Shortest Processing Time first (SPT) rule. The result is stated as follows. Theorem 2.5. For the problem 1jLEgtp jC max , there exists an optimal schedule in which the jobs are ordered according to the SPT rule. Proof. By pairwise job interchange argument. Suppose that there exists an optimal schedule S ¼ ðp1 J j J i p2 Þ with pj > pi , where p1 and p2 denote the partial sequences of S. Let S0 be a schedule with jobs J i and J j of S mutually exchanged, that is, S0 ¼ ðp1 J i J j p2 Þ. In addition, we assume that there are r 1 jobs in p1 and let C l ðSÞ and C l ðS0 Þ denote the complete times of job J l in S and S0 , respectively. We will show that the interchange of jobs J i and J j does not increase the objective value. The repeated implementation of this argument will lead to the optimality of the SPT rule for the problem 1jLEgtp jC max . SpeP 0 cifically, it suffices to show that C j ðS0 Þ 6 C i ðSÞ. To further simplify the notation, let B ¼ r1 k¼1 p½k and let B denote the comple0 tion time of the last job in p1 . Then from Eq. (6) the completion times of job J j in S and job J i in S are
C j ðS0 Þ ¼ B0 þ pi f ðBÞgðrÞ þ pj f ðB þ pi Þgðr þ 1Þ and
C i ðSÞ ¼ B0 þ pj f ðBÞgðrÞ þ pi f B þ pj gðr þ 1Þ; respectively. Thus we have
C j ðS0 Þ C i ðSÞ ¼ pi f ðBÞgðrÞ þ pj f ðB þ pi Þgðr þ 1Þ pj f ðBÞgðrÞ pi f B þ pj gðr þ 1Þ ¼ ðpi pj Þf ðBÞgðrÞ þ pj f ðB þ pi Þgðr þ 1Þ pi f B þ pj gðr þ 1Þ gðr þ 1Þ pj pj gðr þ 1Þ f B þ pj : ¼ pi gðrÞ 1 f ðBÞ þ f ðB þ pi Þ gðrÞ gðrÞ pi pi p
Let a ¼ pji and k ¼ gðrþ1Þ . Clearly, a P 1 and 0 < k 6 1. Thus, by Lemma 2.2, we have gðrÞ
C j ðS0 Þ C i ðSÞ ¼ pi gðrÞðð1 aÞf ðBÞ þ af ðB þ pi Þk f ðB þ api ÞkÞ 6 0: Consequently, C j ðS0 Þ 6 C i ðSÞ. This completes the proof.
h
Townsend [26] studied a single machine scheduling problem with a quadratic cost function of completion times and P showed that the problem 1jj C 2j can be solved optimally by the SPT rule. In some scheduling situations, it needs to consider a polynomial cost function of degree k. Here we consider the single machine problem to minimize the sum of the kth power P of the completion times under model (6), denoted by 1jLEgtp j C kj . And we show that the SPT rule is still optimal for the probP k lem 1jLEgtp j C j . The result is stated in the following theorem. P Theorem 2.6. For the problem 1jLEgtp j C kj , 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 2.5 and let C ½r ðpÞ denote the completion time of the job scheduled in the rth position in a given sequence p. From Eq. (6) the completion times of job J i in S0 and job J j in S are, respectively
C i ðS0 Þ ¼ B0 þ pi f ðBÞgðrÞ and
C j ðSÞ ¼ B0 þ pj f ðBÞgðrÞ:
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From pi < pj , we get C i ðS0 Þ 6 C j ðSÞ. By Theorem 2.5, we have C j ðS0 Þ 6 C i ðSÞ. Thus, for any 1 6 r 6 n, we have C ½r ðS0 Þ 6 C ½r ðSÞ. P k 0 P k P k P k 0 C ½r ðS Þ 6 C ½r ðSÞ ¼ C j ðSÞ. This completes the proof. h Since k is a positive real number, we have C j ðS Þ ¼ As a direct consequence of Theorem 2.6, we have the following corollary. P Corollary 2.7. For the total completion time minimization problem 1jLEgtp j C j , there exists an optimal schedule in which the jobs are ordered according to the SPT rule. For the objective functions of minimizing the weighted sum of completion times and minimizing the maximum lateness, Mosheiov [20] showed that polynomial optimal solutions of the classical version do not hold under model (1). However, under certain conditions the problems can be solved in polynomial time under model (6). P Theorem 2.8. For the problem 1jLEgtp j wj C j , if jobs have reversely agreeable weights, that is, pi 6 pj implies wi P wj for all jobs p J i and J j , then there exists an optimal schedule in which the jobs are ordered in non-decreasing order of wjj (Weighted Shortest Processing Time first rule, WSPT rule). Proof. By pairwise job interchange argument. We still adopt the same notations as in the proof of Theorem 2.5. Suppose that there is an optimal schedule which does not follow the WSPT rule, that is, there exist at least two adjacent jobs, say job J i and p job J j , such that J j is scheduled before J i with wjj > wpii . Note that the weights of the two jobs are reversely agreeable by assumption. Thus, we have pj P pi and wj 6 wi . Hence C j ðS0 Þ 6 C i ðSÞ by Theorem 2.5. Next we will show that the interchange of jobs J i and J j does not increase the objective value. The repeated implementation of this argument will lead to the optimality of the P WSPT rule for the problem 1jLEgtp j wj C j . Specially, it suffices to show that wi C i ðS0 Þ þ wj C j ðS0 Þ 6 wi C i ðSÞ þ wj C j ðSÞ. From Eq. (6), we have
wi C i ðSÞ þ wj C j ðSÞ ¼ wi B0 þ wi pj f ðBÞgðrÞ þ wi pi f B þ pj gðr þ 1Þ þ wj B0 þ wj pj f ðBÞgðrÞ and
wi C i ðS0 Þ þ wj C j ðS0 Þ ¼ wj B0 þ wj pi f ðBÞgðrÞ þ wj pj f ðB þ pi Þgðr þ 1Þ þ wi B0 þ wi pi f ðBÞgðrÞ: Then we have
wi C i ðSÞ þ wj C j ðSÞ wi C i ðS0 Þ wj C j ðS0 Þ
¼ ðwi þ wj Þðpj pi Þf ðBÞgðrÞ þ wi pi f B þ pj gðr þ 1Þ wj pj f ðB þ pi Þgðr þ 1Þ gðr þ 1Þ pj pj wi wj gðr þ 1Þ ¼ ðwi þ wj Þpi gðrÞ 1 f ðBÞ þ f B þ pj f ðB þ pi Þ : gðrÞ gðrÞ pi wi þ wj wi þ wj pi
p
w
j i Let a ¼ pji , k1 ¼ wi þw ; k2 ¼ wiwþw and k ¼ gðrþ1Þ . Clearly a > 1, 0 6 k1 6 k2 6 1 and 0 < k 6 1. Thus, by Lemma 2.4, we have gðrÞ j j
wi C i ðSÞ þ wj C j ðSÞ wi C i ðS0 Þ wj C j ðS0 Þ ¼ ðwi þ wj Þpi gðrÞðða 1Þf ðBÞ þ k2 f ðB þ api Þk k1 af ðB þ pi ÞkÞ P 0: Consequently, wi C i ðS0 Þ þ wj C j ðS0 Þ 6 wi C i ðSÞ þ wj C j ðSÞ. 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. P Corollary 2.9. For the problem 1jLEgtp ; pj ¼ pj wj C j , there exists an optimal schedule in which the jobs are ordered in nonincreasing order of their weights. Let k be a positive real number. If wj pj ¼ k, then jobs have reversely agreeable weights, that is, pi 6 pj implies wi P wj for all jobs J i and J j . Hence, we have the following corollary. P Corollary 2.10. For the problem 1jLEgtp ; wj pj ¼ kj wj C j , where k is a positive real number, there exists an optimal schedule in which the jobs are ordered according to the SPT rule. Next, we will show that the single machine problem to minimize the maximum lateness under model (6), denoted by 1jLEgtp jLmax , 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 J i and J j . Theorem 2.11. For the problem 1jLEgtp jLmax , if the job processing times and the due dates are agreeable, that is, di 6 dj implies pi 6 pj for all jobs J i and J j , then there exists an optimal schedule in which the jobs are ordered in non-decreasing order of dj (i.e. Earliest Due Date first rule, EDD rule). Proof. By pairwise job interchange argument. We still adopt the same notations as in the proof of Theorem 2.5. Suppose that there is an optimal schedule which does not follow the EDD rule, that is, there exist at least two adjacent jobs, say job J i and job J j , such that J j is scheduled before J i with dj > di . Note that the due dates of the two jobs are agreeable by assumption. Thus, we have pj P pi and so C j ðS0 Þ 6 C i ðSÞ by Theorem 2.5. Next we will show that the maximum lateness does not increase by interchanging the jobs J i and J j . The repeated implementation of this argument will lead to the optimality of the EDD rule for the problem 1jLEgtp jLmax . Specially, it suffices to show that maxfLi ðS0 Þ; Lj ðS0 Þg 6 maxfLi ðSÞ; Lj ðSÞg. By definition, the lateness of jobs J i and J j in S and jobs J i and J j in S0 are respectively
Y. Yin et al. / Information Sciences 179 (2009) 2416–2425
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Li ðSÞ ¼ C i ðSÞ di ; Lj ðSÞ ¼ C j ðSÞ dj ; Li ðS0 Þ ¼ C i ðS0 Þ di ; and
Lj ðS0 Þ ¼ C j ðS0 Þ dj : Since dj P di and C j ðS0 Þ 6 C i ðSÞ, we have Lj ðS0 Þ 6 Li ðSÞ and Li ðS0 Þ 6 Li ðSÞ. Hence maxfLi ðS0 Þ; Lj ðS0 Þg 6 maxfLi ðSÞ; Lj ðSÞg. 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 2.12. For the problem 1jLEgtp ; pj ¼ pjLmax , there exists an optimal schedule in which the jobs are ordered according to the EDD rule. Let k be a positive real number. If dj ¼ kpj for all 1 6 j 6 n, then the job processing times and the due dates are agreeable, that is, di 6 dj implies pi 6 pj for all the jobs J i and J j . Hence, we have the following corollary. Corollary 2.13. For the problem 1jLEgtp ; dj ¼ kpj jLmax , where k is a positive real number, there exists an optimal schedule in which 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 the problem reduces to 1jLEgtp ¼ djC max . Hence we have the following corollary. Corollary 2.14. For the problem 1jLEgtp ; dj ¼ djLmax , there exists an optimal schedule in which the jobs are ordered according to the SPT rule.
3. Some m-machine permutation flowshop problems Cheng et al. [9] considered the learning effect on an m-machine flowshop and presented polynomial time optimal solutions for some special cases of the m-machine flowshop problems to minimize makespan and total completion time. In this section, we consider the learning effect in the flowshop environment under the general learning effect model. The flowshop scheduling problem can be described in the following way. There are n jobs J 1 ; J 2 ; . . . ; J n to be processed on m machines M 1 ; M 2 ; . . . ; M m . Each job J j have a due date dj , a positive weight wj and consists of a chain operations O1j ; O2j ; . . . ; Omj , where Oij must be processed on machine M i , i ¼ 1; 2; . . . ; m: Processing of operation Oiþ1;j can start only after Oij has been completed. A machine can handle one job at a time and preemption is not allowed. Moreover, we only consider permutation schedules. If job J j is scheduled in the rth position on machine M i in a processing sequence, then its actual processing time pijr is defined as
pijr ¼ pij f
r1 X
!
pi½l gðrÞ;
r ¼ 1; 2; . . . ; n;
ð7Þ
l¼1
P where 0l¼1 pi½l ¼ 0, pij denotes the normal processing time of job J j on machine M i , pi½l represents the normal processing time of the job scheduled in the lth position on machine M i in the sequence, f : ½0; þ1Þ ! ð0; 1 is a differentiable non-increasing function with f 0 is non-decreasing on ½0; þ1Þ and f ð0Þ ¼ 1, and g : ½1; þ1Þ ! ð0; 1 is a non-increasing function with gð1Þ ¼ 1. For convenience, we still denote the learning effect given in Eq. (7) by the same notation LEgtp as in Section 2. For a given sequence p, C ij ðpÞ represents the completion time of operation Oij , and C j ðpÞ ¼ C mj ðpÞ denotes the completion time of job J j . In the sequel, we consider the special case where the processing times on all the machines for any given job are identical, i.e. pij ¼ pj . For the traditional flowshop problem, Pinedo [24, p. 161] showed that the completion time of the rth job in a given sequence S is
C ½r ðSÞ ¼
r X
p½k þ ðm 1Þ maxfp½k g; 16k6r
k¼1
where p½k is the normal processing time of the job scheduled in the kth position in the sequence. Similarly, the completion time of the rth job in a given sequence S under Eq. (7) is
C ½r ðSÞ ¼
r X k¼1
p½k f
k1 X l¼1
!
(
p½l gðkÞ þ ðm 1Þ max p½k f 16k6r
k1 X
!
)
p½l gðkÞ :
ð8Þ
l¼1
Next, we will show that the SPT rule still provides an optimal schedule for the considered m-machine permutation flowshop makespan problem under model (7), denoted by FmjLEgtp ; prmu; pij ¼ pj jC max . Theorem 3.1. For the problem FmjLEgtp ; prmu; pij ¼ pj jC max , there exists an optimal schedule in which the jobs are ordered according to the SPT rule.
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Proof. By pairwise job interchange argument. Suppose that there exists an optimal schedule S ¼ ðp1 J j J i p2 Þ with pj > pi , where p1 and p2 denote the partial sequences of S. Let S0 be a schedule with jobs J i and J j of S mutually exchanged, that is, S0 ¼ ðp1 J i J j p2 Þ. Furthermore, we assume that there are r 1 jobs in p1 . We will show that the interchange of jobs J i and J j does not increase the objective value. The repeated implementation of this argument will lead to the optimality of the it suffices to show that C j ðS0 Þ 6 C i ðSÞ. To further simplify SPT rule for the problem FmjLEgtp ; prmu; pij ¼ pj jC max . Specifically, P Pr1 Pr1 k1 0 0 the notation, let B ¼ k¼1 p½k and B ¼ k¼1 p½k f l¼1 p½l gðkÞ. Then from Eq. (8), the completion times of job J j in S and job J i in S are
C j ðS0 Þ ¼ B0 þ pi f ðBÞgðrÞ þ pj f ðB þ pi Þgðr þ 1Þ þ ðm 1Þ ! ( ) r2 X p½l gðr 1Þ; pi f ðBÞgðrÞ; pj f ðB þ pi Þgðr þ 1Þ max p½1 ; . . . ; p½r1 f l¼1
and
C i ðSÞ ¼ B0 þ pj f ðBÞgðrÞ þ pi f B þ pj gðr þ 1Þ þ ðm 1Þ ! ( ) r2 X max p½1 ; . . . ; p½r1 f p½l gðr 1Þ; pj f ðBÞgðrÞ; pi f B þ pj gðr þ 1Þ ; l¼1
respectively. Since pi < pj , f and g are non-increasing, we have
pi f ðBÞgðrÞ 6 pj f ðBÞgðrÞ and
pj f ðB þ pi Þgðr þ 1Þ 6 pj f ðBÞgðrÞ: This implies that
( ðm 1Þ max p½1 ; . . . ; p½r1 f ( 6 ðm 1Þ max p½1 ; p½r1 f
r2 X
!
)
p½l gðr 1Þ; pi f ðBÞgðrÞ; pj f ðB þ pi Þgðr þ 1Þ
l¼1 r2 X
! ) p½l gðr 1Þ; . . . ; pj f ðBÞgðrÞ; pi f B þ pj gðr þ 1Þ :
l¼1
In addition, it follows from the proof of Theorem 2.5 that
pi f ðBÞgðrÞ þ pj f ðB þ pi Þgðr þ 1Þ 6 pj f ðBÞgðrÞ þ pi f B þ pj gðr þ 1Þ: Summing up the above arguments, we get C j ðS0 Þ 6 C i ðSÞ. This completes the proof. h Theorem 3.2. For the problem FmjLEgtp ; prmu; pij ¼ pj j which the jobs are ordered according to the SPT rule.
P
C kj , where k is a positive real number, there exists an optimal schedule in
Proof. The proof is similar to those of Theorems 2.6 and 3.1. h As a direct consequence of Theorem 3.2, we have the following corollary. P C j , there exists an optimal schedule in which the jobs are ordered
Corollary 3.3. For the problem FmjLEgtp ; prmu; pij ¼ pj j according to the SPT rule.
The following theorem indicates that it still gives an optimal schedule for the m-machine permutation flowshop total P weighted completion time problem under model (7), denoted by FmjLEgtp ; prmu; pij ¼ pj j wj C j , if jobs have reversely agreeable weights, that is, pi 6 pj implies wi P wj for all jobs J i and J j . P Theorem 3.4. For the problem FmjLEgtp ; prmu; pij ¼ pj j wj C j , if jobs have reversely agreeable weights, that is, pi 6 pj implies wi P wj for all jobs J i and J j , then there exists an optimal schedule in which the jobs are ordered according to the WSPT rule. Proof. By the same pairwise job interchange argument. We still adopt the same notations as in the proof of Theorem 3.1. Suppose that there is an optimal schedule which does not follow the WSPT rule, that is, there exist at least two adjacent jobs, p say job J j and job J i , such that J j is scheduled before J i with wjj > wpii . Note that the weights of the two jobs are reversely agreeable by assumption. Thus, we have pj P pi and wj 6 wi . Hence C j ðS0 Þ 6 C i ðSÞ by Theorem 3.1. Next we will show that the interchange of jobs J i and J j does not increase the objective value. The repeated implementation of this argument will lead P to the optimality of the WSPT rule for the problem FmjLEgtp ; prmu; pij ¼ pj j wj C j . Specially, it suffices to show that 0 0 wi C i ðS Þ þ wj C j ðS Þ 6 wi C i ðSÞ þ wj C j ðSÞ. From Eq. (8), we have
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wi C i ðSÞ þ wj C j ðSÞ ¼ wi B0 þ wi pj f ðBÞgðrÞ þ wi pi f B þ pj gðr þ 1Þ þ wj B0 þ wj pj f ðBÞgðrÞ ! ( ) r2 X p½l gðr 1Þ; pj f ðBÞgðrÞ; pi f B þ pj gðr þ 1Þ þ wi ðm 1Þ max p½1 ; . . . ; p½r1 f l¼1
( þ wj ðm 1Þ max p½1 ; . . . ; p½r1 f
r2 X
!
)
p½l gðr 1Þ; pj f ðBÞgðrÞ
l¼1
and
wi C i ðS0 Þ þ wj C j ðS0 Þ ¼ wj B0 þ wj pi f ðBÞgðrÞ þ wj pj f ðB þ pi Þgðr þ 1Þ þ wi B0 þ wi pi f ðBÞgðrÞ ! ( ) r2 X p½l gðr 1Þ; pi f ðBÞgðrÞ; pj f ðB þ pi Þgðr þ 1Þ þ wj ðm 1Þ max p½1 ; . . . ; p½r1 f l¼1
( þ wi ðm 1Þ max p½1 ; . . . ; p½r1 f
r2 X
!
)
p½l gðr 1Þ; pi f ðBÞgðrÞ :
l¼1
Since pi < pj , f and g are non-increasing, we have
pj f ðBÞgðrÞ P pi f ðBÞf ðrÞ and pj f ðBÞgðrÞ P pj f ðB þ pi Þgðr þ 1Þ: Hence,
( wi ðm 1Þ max p½1 ; . . . ; p½r1 f (
r2 X
!
)
p½l gðr 1Þ; pj f ðBÞgðrÞ; pi f B þ pj gðr þ 1Þ
l¼1
P wi ðm 1Þ max p½1 ; . . . ; p½r1 f
r2 X
!
)
p½l gðr 1Þ; pi f ðBÞgðrÞ
l¼1
and
( wj ðm 1Þ max p½1 ; . . . ; p½r1 f (
r2 X
! p½l gðr 1Þ; pj f ðBÞgðrÞ
l¼1
P wj ðm 1Þ max p½1 ; . . . ; p½r1 f
)
r2 X
!
)
p½l gðr 1Þ; pi f ðBÞgðrÞ; pj f ðB þ pi Þgðr þ 1Þ :
l¼1
On the other hand, it follows from the proof of Theorem 2.8 that
wi B0 þ wi pj f ðBÞgðrÞ þ wi pi f B þ pj gðr þ 1Þ þ wj B þ wj pj f ðBÞgðrÞ P wj B0 þ wj pi f ðBÞgðrÞ þ wj pj f ðB þ pi Þgðr þ 1Þ þ wi B þ wi pi f ðBÞgðrÞ: Summing up the above arguments, we get wi C i ðS0 Þ þ wj C j ðS0 Þ 6 wi C i ðSÞ þ wj C j ðSÞ. This completes the proof. h As a direct consequence of Theorem 3.4, we have the following corollaries. P Corollary 3.5. For the problem FmjLEgtp ; prmu; pij ¼ pj wj C j , there exists an optimal schedule in which the jobs are ordered in non-increasing order of their weights. P Corollary 3.6. For the problem FmjLEgtp ; prmu; pij ¼ pj ; wj pj ¼ kj wj C j , where k is a positive real number, there exists an optimal schedule in which the jobs are ordered according to the SPT rule. Theorem 3.7. For the problem FmjLEgtp ; prmu; pij ¼ pj jLmax , if the job processing times and the due dates are agreeable, that is, di 6 dj implies pi 6 pj for all jobs J i and J j , then there exists an optimal schedule in which the jobs are ordered according to the EDD rule. Proof. The proof is similar to those of Theorems 2.11 and 3.1. h If the normal processing times of all jobs are equal, that is, pij ¼ pj ¼ p for 1 6 i 6 m and 1 6 j 6 n, then we have the following corollary. Corollary 3.8. For the problem FmjLEgtp ; prmu; pij ¼ pj ¼ pjLmax , there exists an optimal schedule in which the jobs are ordered according to the EDD rule. Let k be a positive real number. If pij ¼ pj and 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 J i and J j . Hence, we have the following corollary.
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Corollary 3.9. For the problem FmjLEgtp ; prmu; pij ¼ pj ; dj ¼ kpj jLmax , where k is a positive real number, there exists an optimal schedule in which 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 the problem reduces to FmjLEgtp ; prmu; pij ¼ pj jC max . Hence we have the following corollary. Corollary 3.10. For the problem FmjLEgtp ; prmu; pij ¼ pj ; dj ¼ djLmax , there exists an optimal schedule in which the jobs are ordered according to the SPT rule.
4. Conclusions Scheduling problems with learning effects have recently received growing attention from the scheduling research community. However, most of the research is based on specific learning functions. The aim of the paper is to bring into the scheduling field a general learning effect model where the actual processing time of a job is not only a function of the total normal processing times of the jobs already processed, but also a function of the job’s scheduled position. Particularly, we showed that the single machine makespan and the sum of the kth power of completion times problems remain polynomially solvable under the proposed model. In addition, we showed that the total weighted completion time and the maximum lateness problems are polynomially solvable under certain conditions. Finally, we showed that for some special cases of the m-machine permutation flowshop problems, similar results can also be obtained under the proposed model. These results are significant extensions of some of the existing results on learning effects in the literature. We believe that the general learning effect model offered here will turn out to be more useful in the theory and applications of scheduling. Acknowledgements We express our warmest thanks to Professor Witold Pedrycz, Editor-in-Chief, for communicating the paper, and his useful suggestions. We also express our warmest thanks to the referees for their interest in our work and their value time to read the manuscript very carefully and their valuable comments for improving the paper. This research was supported by the National Natural Science Foundation of China (60774049). References [1] A. Bachman, A. Janiak, Scheduling jobs with position-dependent processing times, Journal of the Operational Research Society 55 (2004) 257–263. [2] A.B. Badiru, Computational survey of univariate and multivariate learning curve models, IEEE Transactions on Engineering Management 39 (1992) 176–188. [3] D. Biskup, Single-machine scheduling with learning considerations, European Journal of Operational Research 115 (1999) 173–178. [4] D. Biskup, A state-of-the-art review on scheduling with learning effects, European Journal of Operational Research 188 (2008) 315–329. [5] D. Biskup, D. Simons, Common due date scheduling with autonomous and induced learning, European Journal of Operational Research 159 (2004) 606– 616. [6] P. Chen, C.-C. Wu, W.-C. Lee, A bi-criteria two-machine flowshop scheduling problem with a learning effect, Journal of the Operational Research Society 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, Applied Mathematics and Computation 184 (2007) 945–949. [8] T.C.E. Cheng, G. Wang, Single machine scheduling with learning effect considerations, Annals of Operations Research 98 (2000) 273–290. [9] 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, Information Sciences 178 (2008) 2476–2487. [10] T. Eren, E. Guner, Minimizing total tardiness in a scheduling problem with a learning effect, Applied Mathematical Modelling 31 (2007) 1351–1361. [11] R.L. Graham, E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling: a survey, Annals of Discrete Mathematics 5 (1979) 287–326. [12] Y.M. Jaber, M. Bonney, The economic manufacture/order quantity (EMQ/EOQ) and the learning curve: past, present, and future, International Journal of Production Economics 59 (1999) 93–102. [13] C. Koulamas, G.J. Kyparisis, Single-machine and two-machine flowshop scheduling with general learning functions, European Journal of Operational Research 178 (2007) 402–407. [14] W.-H. Kuo, D.-L. Yang, Minimizing the total completion time in a single machine scheduling problem with a time-dependent learning effect, European Journal of Operational Research 174 (2006) 1184–1190. [15] W.-H. Kuo, D.-L. Yang, Single-machine group scheduling with a time-dependent learning effect, Computers and Operations Research 33 (2006) 2099– 2112. [16] W.-H. Kuo, D.-L. Yang, Single-machine scheduling problems with the time-dependent learning effect, Computers and Mathematics with Applications 53 (2007) 1733–1739. [17] W.-C. Lee, C.-C. Wu, Minimizing total completion time in a two-machine flowshop with a learning effect, International Journal of Production Economics 88 (2004) 85–92. [18] W.-C. Lee, C.-C. Wu, H.-J. Sung, A bi-criterion single machine scheduling problem with learning considerations, Acta Informatica 40 (2004) 303–315. [19] B.M.T. Lin, Complexity results for single machine scheduling with positional learning effects, Journal of the Operational Research Society 58 (2007) 1099–1102. [20] G. Mosheiov, Scheduling problems with a learning effect, European Journal of Operational Research 132 (2001) 687–692. [21] G. Mosheiov, Parallel machine scheduling with a learning effect, Journal of the Operational Research Society 52 (2001) 1165–1169. [22] G. Mosheiov, J.B. Sidney, Scheduling with general job-dependent learning curves, European Journal of Operational Research 147 (2003) 665–670. [23] S.-J. Park, K.-H. Cho, Real-time preemptive scheduling of sporadic tasks based on supervisory control of discrete event systems, Information Sciences 178 (2008) 3393–3401. [24] M. Pinedo, Scheduling: Theory, Algorithms, and Systems, third ed., Springer, New York, 2008.
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[25] R. Tavakkoli-Moghaddam, A. Rahimi-Vahed, A.H. Mirzaei, A hybrid multi-objective immune algorithm for a flow shop scheduling problem with biobjectives: weighted mean completion time and weighted mean tardiness, Information Sciences 177 (2007) 5072–5090. [26] W. Townsend, The single machine problem with quadratic penalty function of completion times: a branch-and-bound solution, Management Science 24 (1978) 530–533. [27] J.-B. Wang, A note on scheduling problems with learning effects and deteriorating jobs, International Journal of Systems Science 37 (2006) 827–832. [28] J.-B. Wang, Single-machine scheduling problems with the effects of learning and deterioration, Omega 35 (2007) 397–402. [29] J.-B. Wang, C.T. Ng, T.C.E. Cheng, L.L. Liu, Single-machine scheduling with a time-dependent learning effect, International Journal of Production Economics 111 (2008) 802–811. [30] J.-B. Wang, Z.-Q. Xia, Flow-shop scheduling with a learning effect, Journal of the Operational Research Society 56 (2005) 1325–1330. [31] X. Wang, T.C.E. Cheng, Single-machine scheduling with deteriorating jobs and learning effects to minimize the makespan, European Journal of Operational Research 178 (2007) 57–70. [32] C.-C. Wu, W.-C. Lee, Single-machine scheduling problems with a learning effect, Applied Mathematical Modeling 32 (2008) 1191–1197. [33] C.-C. Wu, W.-C. Lee, T. Chen, Heuristic algorithms for solving the maximum lateness scheduling problem with learning considerations, Computers and Industrial Engineering 52 (2007) 124–132. [34] C.-C. Wu, W.-C. Lee, W.-C. Wang, A two-machine flowshop maximum tardiness scheduling problem with a learning effect, International Journal of Advanced Manufacturing Technology 31 (2007) 743–750. [35] D. Xu, Z. Cheng, Y. Yin, H. Li, Makespan minimization for two parallel machines scheduling with a periodic availability constraint, Computers and Operations Research 36 (2009) 1809–1812. [36] D. Xu, K. Sun, H. Li, Parallel machine scheduling with almost periodic maintenance and non-preemptive jobs to minimize makespan, Computers and Operations Research 35 (2008) 1344–1349. [37] D. Xu, Y. Yin, H. Li, Scheduling jobs under increasing linear machine maintenance time, Journal of Scheduling, doi:10.1007/s10951-008-0057-9. [38] D. Xu, Y. Yin, H. Li, A note on ‘‘scheduling of nonresumable jobs and flexible maintenance activities on a single machine to minimize makespan, European Journal of Operational Research 197 (2009) 825–827. [39] C.-L. Zhao, Q.-L. Zhang, H.-Y. Tang, Machine scheduling problems with learning effects, Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis 11 (2004) 741–750.
Contents lists available at ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins
Some scheduling problems with general position-dependent and time-dependent learning effects Yunqiang Yin a, Dehua Xu a, Kaibiao Sun b, Hongxing Li b,* a b
College of Mathematics and Information Sciences, East China Institute of Technology, Fuzhou, Jiangxi 344000, China School of Electronic and Information Engineering, Dalian University of Technology, Dalian, Liaoning 116024, China
a r t i c l e
i n f o
Article history: Received 21 April 2008 Received in revised form 11 January 2009 Accepted 26 February 2009
Keywords: Scheduling Learning effect Time-dependent Position-dependent Single-machine Flowshop
a b s t r a c t In scheduling problems with learning effects, most of the research is based on specific learning functions. In this paper, we develop a general model with learning effects where the actual processing time of a job is not only a function of the total normal processing times of the jobs already processed, but also a function of the job’s scheduled position. In particular, it is shown that some single machine scheduling problems and m-machine permutation flowshop problems are still polynomially solvable under the proposed model. These results are significant extensions of some of the existing results on learning effects in the literature. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Scheduling problems have received considerable attention since the middle of the last century. However, most researchers assume that job processing times are known and fixed during the whole processing process (see, e.g., [23,25,35–38]). 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 (see, e.g., [2,3]). During the last few years, learning effect has attracted growing attention in the scheduling community on account of its significance. There have been many attempts to formulate learning effect in a quantitative form as a function of learning variables, called a learning curve. Most of the concepts assume that the learning curve is a non-increasing function which depends on the jobs already performed. For a survey on learning curves, the reader is refereed to Jaber and Bonney [12]. To the best of our knowledge, Biskup [3] and Cheng and Wang [8] are among the pioneers that brought the concept of learning effect into the field of scheduling. Since then, scheduling problems with learning effects have received considerable attention. According to a recent state-of-the-art literature review paper of Biskup [4], over ten learning effect models have been proposed and investigated since 1999. One can distinguish these models into three main categories, that is, positiondependent (see, e.g., [1,3,5,8,22,30]), time-dependent (see, e.g., [13,14]) and a combination of the two (see, e.g., [9,32]). In what follow, we will briefly review some of these models. In the classical position-dependent learning effect model (Biskup [3]), the actual processing time pir of job J i when it is scheduled in the rth position in a processing sequence is defined as * Corresponding author. Tel.: +86 411 84706402; fax: +86 411 84706405. E-mail addresses: [email protected] (Y. Yin), [email protected] (D. Xu), [email protected] (K. Sun), [email protected] (H. Li). 0020-0255/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2009.02.015
Y. Yin et al. / Information Sciences 179 (2009) 2416–2425
pir ¼ pi r a ;
2417
ð1Þ
where a is a non-positive learning index and pi denotes the normal processing time of job J i . Biskup [3] showed that single machine scheduling problems under model (1) are polynomially solvable if the objective is to minimize the deviation from a common due date or to minimize the sum of flow times. Many researchers have studied such a learning effect model and its variants thereafter, a sample of these papers include [1,6–8,10,16–21,27,28,30,31,33,34,39]. Note that position-dependent learning effects neglect the processing times of the jobs already processed. If human interactions have significant impacts during the processing of jobs, the processing times will be added to the employees’ experience and thus cause learning effects. For situations like this it might be more appropriate to consider a time-dependent learning effect [4]. Kuo and Yang [14] proposed the following time-dependent learning effect model:
pir ¼ pi 1 þ
!a
r1 X
p½k
ð2Þ
;
k¼1
where a 6 0 denotes the learning index and p½k represents the normal processing time of a job when it is scheduled in the kth position in the sequence. Kuo and Yang [14] showed that the single machine scheduling problem to minimize total complete time is polynomially solvable under model (2). For more papers about this time-dependent learning effect model, the reader is refereed to [15,16,29]. Besides, Koulamas and Kyparisis [13] considered the following time-dependent learning effect model:
Pr1
k¼1 p½k
!a
pir ¼ pi 1 Pn
ð3Þ
;
k¼1 pk
where a P 1 denotes the learning index. They showed that the single machine makespan and total completion time minimization problems under model (3) are polynomially solvable. Recently, Wu and Lee [32] 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 can be described as follows:
Pr1
k¼1 p½k Pn k¼1 pk
pir ¼ pi 1 þ
!a 1 r a2 ;
ð4Þ
where a1 < 0 and a2 < 0 denote two learning indices. They showed that the single machine makespan and the total completion time problems are polynomially solvable under model (4) and that the total weighted completion time has a polynomial optimal solution under certain agreeable conditions. Cheng et al. [9] consider a similar model that can be described as follows:
Pr1 pir ¼ pi 1
p½k Pk¼1 n k¼1 pk
!a 1 r a2 ;
ð5Þ
where a1 and a2 denote two learning indices with a1 P 1 and a2 < 0. They obtained similar results as in Wu and Lee [32]. In addition, they presented polynomial optimal solutions for some special cases of the m-machine flowshop problems to minimize makespan and total completion time. Note that all of the papers mentioned above concerning leaning effects on scheduling are based on specific a problems P a and pir ¼ pi 1 Px 1 r a2 , where x ¼ r1 learning functions, such as pir ¼ pi r a , pir ¼ pi ð1 þ xÞa , pir ¼ pi 1 Px i¼1 p½i and Pn P ¼ i¼1 pi . It is natural to ask whether the above results are still true for a general learning effect model of the form pir ¼ pi f ðxÞgðrÞ, where f and g are two real functions. This paper attempts to answer this question. We show that some single machine scheduling problems and m-machine permutation flowshop problems are still polynomially solvable under the above learning effect model if f : ½0; þ1Þ ! ð0; 1 is a differentiable non-increasing function with f 0 is non-decreasing on ½0; þ1Þ and f ð0Þ ¼ 1, and g : ½1; þ1Þ ! ð0; 1 is a non-increasing function with gð1Þ ¼ 1. These results are significant extensions of some of the existing results on learning effects in the literature. 2. Some single machine problems Assume that there are n jobs J 1 ; J 2 ; . . . ; J n to be processed on a single machine. The machine can handle one job at a time and preemption is not allowed. Each job J j has a normal processing time pj , a due date dj and a positive weight wj . If job J j is scheduled in the rth position in a processing sequence, then its actual processing time is defined as
pjr ¼ pj f
r1 X
! p½k gðrÞ;
r ¼ 1; 2; . . . ; n;
ð6Þ
k¼1
P0 where k¼1 p½k ¼ 0, p½k denotes the normal processing time of the job scheduled in the kth position in the sequence, f : ½0; þ1Þ ! ð0; 1 is a differentiable non-increasing function with f 0 is non-decreasing on ½0; þ1Þ and f ð0Þ ¼ 1, and
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g : ½1; þ1Þ ! ð0; 1 is a non-increasing function with gð1Þ ¼ 1. For convenience, we denote the learning effect given in Eq. (6) by LEgtp . In such a learning effect model, we can see that the actual processing time of a job depends not only on the total processing time of the jobs already processed, but also on its scheduled position. It is also evident from the model that the longer the already processed jobs or the later the job position is, the stronger the learning effect is on the subsequent jobs that are yet to be processed. Remark 2.1. One may note that model (6) is a generalization of models 1–5. Indeed: (i) (ii) (iii) (iv)
Define f : ½0; þ1Þ ! ð0; 1 by f ðxÞ ¼ 1. Then model (6) is a generalization of model (1). Define g : ½1; of models (2) and (3). þ1Þ a ! ð0; 1 by gðxÞ ¼ 1. Then model (6) is a generalization P Let f ðxÞ ¼ 1 þ Px 1 and gðxÞ ¼ xa2 , where a1 < 0, a2 < 0 and P ¼ nk¼1 pk . Then model (6) reduces to model (4). P a Let f ðxÞ ¼ 1 Px 1 and gðxÞ ¼ xa2 , where a1 P 1, a2 < 0 and P ¼ nk¼1 pk . Then model (6) reduces to model (5).
In this section we will study some single machine problems under model (6). Before developing the results, we first give some lemmas which are useful for the following theorems. Lemma 2.2. If f : ½0; þ1Þ ! ð0; 1 is a differentiable non-increasing function and its derivative f 0 is non-decreasing on ½0; þ1Þ, then
ð1 aÞf ðaÞ þ akf ða þ tÞ kf ða þ atÞ 6 0 for a P 1; 0 < k 6 1; a P 0 and t P 0. Proof. Let
FðtÞ ¼ ð1 aÞf ðaÞ þ akf ða þ tÞ kf ða þ atÞ: Taking the first derivative of FðtÞ with respect to t, we have
F 0 ðtÞ ¼ akf 0 ða þ tÞ akf 0 ða þ atÞ ¼ akðf 0 ða þ tÞ f 0 ða þ atÞÞ: Since a P 1, t P 0 and f 0 is non-decreasing, we have f 0 ða þ tÞ f 0 ða þ atÞ 6 0 and so F 0 ðtÞ 6 0. This implies that FðtÞ is nonincreasing on t P 0. Since a P 1; 0 < k 6 1 and f ðaÞ P 0, we have
FðtÞ 6 Fð0Þ ¼ ð1 aÞf ðaÞ þ akf ðaÞ kf ðaÞ ¼ ð1 aÞð1 kÞf ðaÞ 6 0: This completes the proof. h Lemma 2.3. If f : ½0; þ1Þ ! ð0; 1 is a differentiable non-increasing function and its derivative f 0 is non-decreasing on ½0; þ1Þ, then 0
f ðaÞ þ kk2 tf ða þ tÞ kk1 f ða þ tÞ P 0 for 0 < k 6 1, 0 6 k1 6 k2 6 1; a P 0 and t P 0. Proof. Let 0
Fðk1 ; k2 ; tÞ ¼ f ðaÞ þ kk2 tf ða þ tÞ kk1 f ða þ tÞ: Then
oF ok1
¼ kf ða þ tÞ 6 0,
oF ok2
0
¼ ktf ða þ tÞ 6 0. Since 0 6 k1 6 k2 6 1, we have 0
Fðk1 ; k2 ; tÞ P Fð1; 1; tÞ ¼ f ðaÞ þ ktf ða þ tÞ kf ða þ tÞ: By Lagrange mean value theorem, there exists a value n between a and a þ t such that
f ða þ tÞ f ðaÞ ¼ f 0 ðnÞt: Since 0 < k 6 1; t P 0 and f 0 is non-decreasing, we have 0
0
Fðk1 ; k2 ; tÞ P f ðaÞ þ ktf ða þ tÞ kf ða þ tÞ P kf ðaÞ þ ktf ða þ tÞ kf ða þ tÞ ¼ ktðf 0 ða þ tÞ f 0 ðnÞÞ P 0: This completes the proof. h Lemma 2.4. If f : ½0; þ1Þ ! ð0; 1 is a differentiable non-increasing function and its derivative f 0 is non-decreasing on ½0; þ1Þ, then
ða 1Þf ðaÞ þ kk2 f ða þ atÞ akk1 f ða þ tÞ P 0 for a P 1; 0 < k 6 1, 0 6 k1 6 k2 6 1; a P 0 and t P 0.
Y. Yin et al. / Information Sciences 179 (2009) 2416–2425
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Proof. Let
FðaÞ ¼ ða 1Þf ðaÞ þ kk2 f ða þ atÞ akk1 f ða þ tÞ: Taking the first derivative of FðaÞ with respect to a, we have 0
F 0 ðaÞ ¼ f ðaÞ þ kk2 tf ða þ atÞ kk1 f ða þ tÞ: Since a P 1, t P 0 and f 0 is non-decreasing, by Lemma 2.3, we have 0
F 0 ðaÞ P F 0 ð1Þ ¼ f ðaÞ þ kk2 tf ða þ tÞ kk1 f ða þ tÞ P 0: This implies that FðaÞ is non-decreasing on a P 1 and so
FðaÞ P Fð1Þ ¼ kk2 f ða þ tÞ kk1 f ða þ tÞ ¼ kf ða þ tÞðk2 k1 Þ P 0: This completes the proof. h In the sequel, we will adopt the three-field notation scheme ajbjc introduced by Graham et al. [11] to denote the considered problems in this paper. First, we study the single machine makespan minimization problem under model (6), denoted by 1jLEgtp jC max . And we show that an optimal schedule for the problem can be obtained by the Shortest Processing Time first (SPT) rule. The result is stated as follows. Theorem 2.5. For the problem 1jLEgtp jC max , there exists an optimal schedule in which the jobs are ordered according to the SPT rule. Proof. By pairwise job interchange argument. Suppose that there exists an optimal schedule S ¼ ðp1 J j J i p2 Þ with pj > pi , where p1 and p2 denote the partial sequences of S. Let S0 be a schedule with jobs J i and J j of S mutually exchanged, that is, S0 ¼ ðp1 J i J j p2 Þ. In addition, we assume that there are r 1 jobs in p1 and let C l ðSÞ and C l ðS0 Þ denote the complete times of job J l in S and S0 , respectively. We will show that the interchange of jobs J i and J j does not increase the objective value. The repeated implementation of this argument will lead to the optimality of the SPT rule for the problem 1jLEgtp jC max . SpeP 0 cifically, it suffices to show that C j ðS0 Þ 6 C i ðSÞ. To further simplify the notation, let B ¼ r1 k¼1 p½k and let B denote the comple0 tion time of the last job in p1 . Then from Eq. (6) the completion times of job J j in S and job J i in S are
C j ðS0 Þ ¼ B0 þ pi f ðBÞgðrÞ þ pj f ðB þ pi Þgðr þ 1Þ and
C i ðSÞ ¼ B0 þ pj f ðBÞgðrÞ þ pi f B þ pj gðr þ 1Þ; respectively. Thus we have
C j ðS0 Þ C i ðSÞ ¼ pi f ðBÞgðrÞ þ pj f ðB þ pi Þgðr þ 1Þ pj f ðBÞgðrÞ pi f B þ pj gðr þ 1Þ ¼ ðpi pj Þf ðBÞgðrÞ þ pj f ðB þ pi Þgðr þ 1Þ pi f B þ pj gðr þ 1Þ gðr þ 1Þ pj pj gðr þ 1Þ f B þ pj : ¼ pi gðrÞ 1 f ðBÞ þ f ðB þ pi Þ gðrÞ gðrÞ pi pi p
Let a ¼ pji and k ¼ gðrþ1Þ . Clearly, a P 1 and 0 < k 6 1. Thus, by Lemma 2.2, we have gðrÞ
C j ðS0 Þ C i ðSÞ ¼ pi gðrÞðð1 aÞf ðBÞ þ af ðB þ pi Þk f ðB þ api ÞkÞ 6 0: Consequently, C j ðS0 Þ 6 C i ðSÞ. This completes the proof.
h
Townsend [26] studied a single machine scheduling problem with a quadratic cost function of completion times and P showed that the problem 1jj C 2j can be solved optimally by the SPT rule. In some scheduling situations, it needs to consider a polynomial cost function of degree k. Here we consider the single machine problem to minimize the sum of the kth power P of the completion times under model (6), denoted by 1jLEgtp j C kj . And we show that the SPT rule is still optimal for the probP k lem 1jLEgtp j C j . The result is stated in the following theorem. P Theorem 2.6. For the problem 1jLEgtp j C kj , 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 2.5 and let C ½r ðpÞ denote the completion time of the job scheduled in the rth position in a given sequence p. From Eq. (6) the completion times of job J i in S0 and job J j in S are, respectively
C i ðS0 Þ ¼ B0 þ pi f ðBÞgðrÞ and
C j ðSÞ ¼ B0 þ pj f ðBÞgðrÞ:
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From pi < pj , we get C i ðS0 Þ 6 C j ðSÞ. By Theorem 2.5, we have C j ðS0 Þ 6 C i ðSÞ. Thus, for any 1 6 r 6 n, we have C ½r ðS0 Þ 6 C ½r ðSÞ. P k 0 P k P k P k 0 C ½r ðS Þ 6 C ½r ðSÞ ¼ C j ðSÞ. This completes the proof. h Since k is a positive real number, we have C j ðS Þ ¼ As a direct consequence of Theorem 2.6, we have the following corollary. P Corollary 2.7. For the total completion time minimization problem 1jLEgtp j C j , there exists an optimal schedule in which the jobs are ordered according to the SPT rule. For the objective functions of minimizing the weighted sum of completion times and minimizing the maximum lateness, Mosheiov [20] showed that polynomial optimal solutions of the classical version do not hold under model (1). However, under certain conditions the problems can be solved in polynomial time under model (6). P Theorem 2.8. For the problem 1jLEgtp j wj C j , if jobs have reversely agreeable weights, that is, pi 6 pj implies wi P wj for all jobs p J i and J j , then there exists an optimal schedule in which the jobs are ordered in non-decreasing order of wjj (Weighted Shortest Processing Time first rule, WSPT rule). Proof. By pairwise job interchange argument. We still adopt the same notations as in the proof of Theorem 2.5. Suppose that there is an optimal schedule which does not follow the WSPT rule, that is, there exist at least two adjacent jobs, say job J i and p job J j , such that J j is scheduled before J i with wjj > wpii . Note that the weights of the two jobs are reversely agreeable by assumption. Thus, we have pj P pi and wj 6 wi . Hence C j ðS0 Þ 6 C i ðSÞ by Theorem 2.5. Next we will show that the interchange of jobs J i and J j does not increase the objective value. The repeated implementation of this argument will lead to the optimality of the P WSPT rule for the problem 1jLEgtp j wj C j . Specially, it suffices to show that wi C i ðS0 Þ þ wj C j ðS0 Þ 6 wi C i ðSÞ þ wj C j ðSÞ. From Eq. (6), we have
wi C i ðSÞ þ wj C j ðSÞ ¼ wi B0 þ wi pj f ðBÞgðrÞ þ wi pi f B þ pj gðr þ 1Þ þ wj B0 þ wj pj f ðBÞgðrÞ and
wi C i ðS0 Þ þ wj C j ðS0 Þ ¼ wj B0 þ wj pi f ðBÞgðrÞ þ wj pj f ðB þ pi Þgðr þ 1Þ þ wi B0 þ wi pi f ðBÞgðrÞ: Then we have
wi C i ðSÞ þ wj C j ðSÞ wi C i ðS0 Þ wj C j ðS0 Þ
¼ ðwi þ wj Þðpj pi Þf ðBÞgðrÞ þ wi pi f B þ pj gðr þ 1Þ wj pj f ðB þ pi Þgðr þ 1Þ gðr þ 1Þ pj pj wi wj gðr þ 1Þ ¼ ðwi þ wj Þpi gðrÞ 1 f ðBÞ þ f B þ pj f ðB þ pi Þ : gðrÞ gðrÞ pi wi þ wj wi þ wj pi
p
w
j i Let a ¼ pji , k1 ¼ wi þw ; k2 ¼ wiwþw and k ¼ gðrþ1Þ . Clearly a > 1, 0 6 k1 6 k2 6 1 and 0 < k 6 1. Thus, by Lemma 2.4, we have gðrÞ j j
wi C i ðSÞ þ wj C j ðSÞ wi C i ðS0 Þ wj C j ðS0 Þ ¼ ðwi þ wj Þpi gðrÞðða 1Þf ðBÞ þ k2 f ðB þ api Þk k1 af ðB þ pi ÞkÞ P 0: Consequently, wi C i ðS0 Þ þ wj C j ðS0 Þ 6 wi C i ðSÞ þ wj C j ðSÞ. 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. P Corollary 2.9. For the problem 1jLEgtp ; pj ¼ pj wj C j , there exists an optimal schedule in which the jobs are ordered in nonincreasing order of their weights. Let k be a positive real number. If wj pj ¼ k, then jobs have reversely agreeable weights, that is, pi 6 pj implies wi P wj for all jobs J i and J j . Hence, we have the following corollary. P Corollary 2.10. For the problem 1jLEgtp ; wj pj ¼ kj wj C j , where k is a positive real number, there exists an optimal schedule in which the jobs are ordered according to the SPT rule. Next, we will show that the single machine problem to minimize the maximum lateness under model (6), denoted by 1jLEgtp jLmax , 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 J i and J j . Theorem 2.11. For the problem 1jLEgtp jLmax , if the job processing times and the due dates are agreeable, that is, di 6 dj implies pi 6 pj for all jobs J i and J j , then there exists an optimal schedule in which the jobs are ordered in non-decreasing order of dj (i.e. Earliest Due Date first rule, EDD rule). Proof. By pairwise job interchange argument. We still adopt the same notations as in the proof of Theorem 2.5. Suppose that there is an optimal schedule which does not follow the EDD rule, that is, there exist at least two adjacent jobs, say job J i and job J j , such that J j is scheduled before J i with dj > di . Note that the due dates of the two jobs are agreeable by assumption. Thus, we have pj P pi and so C j ðS0 Þ 6 C i ðSÞ by Theorem 2.5. Next we will show that the maximum lateness does not increase by interchanging the jobs J i and J j . The repeated implementation of this argument will lead to the optimality of the EDD rule for the problem 1jLEgtp jLmax . Specially, it suffices to show that maxfLi ðS0 Þ; Lj ðS0 Þg 6 maxfLi ðSÞ; Lj ðSÞg. By definition, the lateness of jobs J i and J j in S and jobs J i and J j in S0 are respectively
Y. Yin et al. / Information Sciences 179 (2009) 2416–2425
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Li ðSÞ ¼ C i ðSÞ di ; Lj ðSÞ ¼ C j ðSÞ dj ; Li ðS0 Þ ¼ C i ðS0 Þ di ; and
Lj ðS0 Þ ¼ C j ðS0 Þ dj : Since dj P di and C j ðS0 Þ 6 C i ðSÞ, we have Lj ðS0 Þ 6 Li ðSÞ and Li ðS0 Þ 6 Li ðSÞ. Hence maxfLi ðS0 Þ; Lj ðS0 Þg 6 maxfLi ðSÞ; Lj ðSÞg. 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 2.12. For the problem 1jLEgtp ; pj ¼ pjLmax , there exists an optimal schedule in which the jobs are ordered according to the EDD rule. Let k be a positive real number. If dj ¼ kpj for all 1 6 j 6 n, then the job processing times and the due dates are agreeable, that is, di 6 dj implies pi 6 pj for all the jobs J i and J j . Hence, we have the following corollary. Corollary 2.13. For the problem 1jLEgtp ; dj ¼ kpj jLmax , where k is a positive real number, there exists an optimal schedule in which 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 the problem reduces to 1jLEgtp ¼ djC max . Hence we have the following corollary. Corollary 2.14. For the problem 1jLEgtp ; dj ¼ djLmax , there exists an optimal schedule in which the jobs are ordered according to the SPT rule.
3. Some m-machine permutation flowshop problems Cheng et al. [9] considered the learning effect on an m-machine flowshop and presented polynomial time optimal solutions for some special cases of the m-machine flowshop problems to minimize makespan and total completion time. In this section, we consider the learning effect in the flowshop environment under the general learning effect model. The flowshop scheduling problem can be described in the following way. There are n jobs J 1 ; J 2 ; . . . ; J n to be processed on m machines M 1 ; M 2 ; . . . ; M m . Each job J j have a due date dj , a positive weight wj and consists of a chain operations O1j ; O2j ; . . . ; Omj , where Oij must be processed on machine M i , i ¼ 1; 2; . . . ; m: Processing of operation Oiþ1;j can start only after Oij has been completed. A machine can handle one job at a time and preemption is not allowed. Moreover, we only consider permutation schedules. If job J j is scheduled in the rth position on machine M i in a processing sequence, then its actual processing time pijr is defined as
pijr ¼ pij f
r1 X
!
pi½l gðrÞ;
r ¼ 1; 2; . . . ; n;
ð7Þ
l¼1
P where 0l¼1 pi½l ¼ 0, pij denotes the normal processing time of job J j on machine M i , pi½l represents the normal processing time of the job scheduled in the lth position on machine M i in the sequence, f : ½0; þ1Þ ! ð0; 1 is a differentiable non-increasing function with f 0 is non-decreasing on ½0; þ1Þ and f ð0Þ ¼ 1, and g : ½1; þ1Þ ! ð0; 1 is a non-increasing function with gð1Þ ¼ 1. For convenience, we still denote the learning effect given in Eq. (7) by the same notation LEgtp as in Section 2. For a given sequence p, C ij ðpÞ represents the completion time of operation Oij , and C j ðpÞ ¼ C mj ðpÞ denotes the completion time of job J j . In the sequel, we consider the special case where the processing times on all the machines for any given job are identical, i.e. pij ¼ pj . For the traditional flowshop problem, Pinedo [24, p. 161] showed that the completion time of the rth job in a given sequence S is
C ½r ðSÞ ¼
r X
p½k þ ðm 1Þ maxfp½k g; 16k6r
k¼1
where p½k is the normal processing time of the job scheduled in the kth position in the sequence. Similarly, the completion time of the rth job in a given sequence S under Eq. (7) is
C ½r ðSÞ ¼
r X k¼1
p½k f
k1 X l¼1
!
(
p½l gðkÞ þ ðm 1Þ max p½k f 16k6r
k1 X
!
)
p½l gðkÞ :
ð8Þ
l¼1
Next, we will show that the SPT rule still provides an optimal schedule for the considered m-machine permutation flowshop makespan problem under model (7), denoted by FmjLEgtp ; prmu; pij ¼ pj jC max . Theorem 3.1. For the problem FmjLEgtp ; prmu; pij ¼ pj jC max , there exists an optimal schedule in which the jobs are ordered according to the SPT rule.
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Proof. By pairwise job interchange argument. Suppose that there exists an optimal schedule S ¼ ðp1 J j J i p2 Þ with pj > pi , where p1 and p2 denote the partial sequences of S. Let S0 be a schedule with jobs J i and J j of S mutually exchanged, that is, S0 ¼ ðp1 J i J j p2 Þ. Furthermore, we assume that there are r 1 jobs in p1 . We will show that the interchange of jobs J i and J j does not increase the objective value. The repeated implementation of this argument will lead to the optimality of the it suffices to show that C j ðS0 Þ 6 C i ðSÞ. To further simplify SPT rule for the problem FmjLEgtp ; prmu; pij ¼ pj jC max . Specifically, P Pr1 Pr1 k1 0 0 the notation, let B ¼ k¼1 p½k and B ¼ k¼1 p½k f l¼1 p½l gðkÞ. Then from Eq. (8), the completion times of job J j in S and job J i in S are
C j ðS0 Þ ¼ B0 þ pi f ðBÞgðrÞ þ pj f ðB þ pi Þgðr þ 1Þ þ ðm 1Þ ! ( ) r2 X p½l gðr 1Þ; pi f ðBÞgðrÞ; pj f ðB þ pi Þgðr þ 1Þ max p½1 ; . . . ; p½r1 f l¼1
and
C i ðSÞ ¼ B0 þ pj f ðBÞgðrÞ þ pi f B þ pj gðr þ 1Þ þ ðm 1Þ ! ( ) r2 X max p½1 ; . . . ; p½r1 f p½l gðr 1Þ; pj f ðBÞgðrÞ; pi f B þ pj gðr þ 1Þ ; l¼1
respectively. Since pi < pj , f and g are non-increasing, we have
pi f ðBÞgðrÞ 6 pj f ðBÞgðrÞ and
pj f ðB þ pi Þgðr þ 1Þ 6 pj f ðBÞgðrÞ: This implies that
( ðm 1Þ max p½1 ; . . . ; p½r1 f ( 6 ðm 1Þ max p½1 ; p½r1 f
r2 X
!
)
p½l gðr 1Þ; pi f ðBÞgðrÞ; pj f ðB þ pi Þgðr þ 1Þ
l¼1 r2 X
! ) p½l gðr 1Þ; . . . ; pj f ðBÞgðrÞ; pi f B þ pj gðr þ 1Þ :
l¼1
In addition, it follows from the proof of Theorem 2.5 that
pi f ðBÞgðrÞ þ pj f ðB þ pi Þgðr þ 1Þ 6 pj f ðBÞgðrÞ þ pi f B þ pj gðr þ 1Þ: Summing up the above arguments, we get C j ðS0 Þ 6 C i ðSÞ. This completes the proof. h Theorem 3.2. For the problem FmjLEgtp ; prmu; pij ¼ pj j which the jobs are ordered according to the SPT rule.
P
C kj , where k is a positive real number, there exists an optimal schedule in
Proof. The proof is similar to those of Theorems 2.6 and 3.1. h As a direct consequence of Theorem 3.2, we have the following corollary. P C j , there exists an optimal schedule in which the jobs are ordered
Corollary 3.3. For the problem FmjLEgtp ; prmu; pij ¼ pj j according to the SPT rule.
The following theorem indicates that it still gives an optimal schedule for the m-machine permutation flowshop total P weighted completion time problem under model (7), denoted by FmjLEgtp ; prmu; pij ¼ pj j wj C j , if jobs have reversely agreeable weights, that is, pi 6 pj implies wi P wj for all jobs J i and J j . P Theorem 3.4. For the problem FmjLEgtp ; prmu; pij ¼ pj j wj C j , if jobs have reversely agreeable weights, that is, pi 6 pj implies wi P wj for all jobs J i and J j , then there exists an optimal schedule in which the jobs are ordered according to the WSPT rule. Proof. By the same pairwise job interchange argument. We still adopt the same notations as in the proof of Theorem 3.1. Suppose that there is an optimal schedule which does not follow the WSPT rule, that is, there exist at least two adjacent jobs, p say job J j and job J i , such that J j is scheduled before J i with wjj > wpii . Note that the weights of the two jobs are reversely agreeable by assumption. Thus, we have pj P pi and wj 6 wi . Hence C j ðS0 Þ 6 C i ðSÞ by Theorem 3.1. Next we will show that the interchange of jobs J i and J j does not increase the objective value. The repeated implementation of this argument will lead P to the optimality of the WSPT rule for the problem FmjLEgtp ; prmu; pij ¼ pj j wj C j . Specially, it suffices to show that 0 0 wi C i ðS Þ þ wj C j ðS Þ 6 wi C i ðSÞ þ wj C j ðSÞ. From Eq. (8), we have
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wi C i ðSÞ þ wj C j ðSÞ ¼ wi B0 þ wi pj f ðBÞgðrÞ þ wi pi f B þ pj gðr þ 1Þ þ wj B0 þ wj pj f ðBÞgðrÞ ! ( ) r2 X p½l gðr 1Þ; pj f ðBÞgðrÞ; pi f B þ pj gðr þ 1Þ þ wi ðm 1Þ max p½1 ; . . . ; p½r1 f l¼1
( þ wj ðm 1Þ max p½1 ; . . . ; p½r1 f
r2 X
!
)
p½l gðr 1Þ; pj f ðBÞgðrÞ
l¼1
and
wi C i ðS0 Þ þ wj C j ðS0 Þ ¼ wj B0 þ wj pi f ðBÞgðrÞ þ wj pj f ðB þ pi Þgðr þ 1Þ þ wi B0 þ wi pi f ðBÞgðrÞ ! ( ) r2 X p½l gðr 1Þ; pi f ðBÞgðrÞ; pj f ðB þ pi Þgðr þ 1Þ þ wj ðm 1Þ max p½1 ; . . . ; p½r1 f l¼1
( þ wi ðm 1Þ max p½1 ; . . . ; p½r1 f
r2 X
!
)
p½l gðr 1Þ; pi f ðBÞgðrÞ :
l¼1
Since pi < pj , f and g are non-increasing, we have
pj f ðBÞgðrÞ P pi f ðBÞf ðrÞ and pj f ðBÞgðrÞ P pj f ðB þ pi Þgðr þ 1Þ: Hence,
( wi ðm 1Þ max p½1 ; . . . ; p½r1 f (
r2 X
!
)
p½l gðr 1Þ; pj f ðBÞgðrÞ; pi f B þ pj gðr þ 1Þ
l¼1
P wi ðm 1Þ max p½1 ; . . . ; p½r1 f
r2 X
!
)
p½l gðr 1Þ; pi f ðBÞgðrÞ
l¼1
and
( wj ðm 1Þ max p½1 ; . . . ; p½r1 f (
r2 X
! p½l gðr 1Þ; pj f ðBÞgðrÞ
l¼1
P wj ðm 1Þ max p½1 ; . . . ; p½r1 f
)
r2 X
!
)
p½l gðr 1Þ; pi f ðBÞgðrÞ; pj f ðB þ pi Þgðr þ 1Þ :
l¼1
On the other hand, it follows from the proof of Theorem 2.8 that
wi B0 þ wi pj f ðBÞgðrÞ þ wi pi f B þ pj gðr þ 1Þ þ wj B þ wj pj f ðBÞgðrÞ P wj B0 þ wj pi f ðBÞgðrÞ þ wj pj f ðB þ pi Þgðr þ 1Þ þ wi B þ wi pi f ðBÞgðrÞ: Summing up the above arguments, we get wi C i ðS0 Þ þ wj C j ðS0 Þ 6 wi C i ðSÞ þ wj C j ðSÞ. This completes the proof. h As a direct consequence of Theorem 3.4, we have the following corollaries. P Corollary 3.5. For the problem FmjLEgtp ; prmu; pij ¼ pj wj C j , there exists an optimal schedule in which the jobs are ordered in non-increasing order of their weights. P Corollary 3.6. For the problem FmjLEgtp ; prmu; pij ¼ pj ; wj pj ¼ kj wj C j , where k is a positive real number, there exists an optimal schedule in which the jobs are ordered according to the SPT rule. Theorem 3.7. For the problem FmjLEgtp ; prmu; pij ¼ pj jLmax , if the job processing times and the due dates are agreeable, that is, di 6 dj implies pi 6 pj for all jobs J i and J j , then there exists an optimal schedule in which the jobs are ordered according to the EDD rule. Proof. The proof is similar to those of Theorems 2.11 and 3.1. h If the normal processing times of all jobs are equal, that is, pij ¼ pj ¼ p for 1 6 i 6 m and 1 6 j 6 n, then we have the following corollary. Corollary 3.8. For the problem FmjLEgtp ; prmu; pij ¼ pj ¼ pjLmax , there exists an optimal schedule in which the jobs are ordered according to the EDD rule. Let k be a positive real number. If pij ¼ pj and 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 J i and J j . Hence, we have the following corollary.
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Corollary 3.9. For the problem FmjLEgtp ; prmu; pij ¼ pj ; dj ¼ kpj jLmax , where k is a positive real number, there exists an optimal schedule in which 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 the problem reduces to FmjLEgtp ; prmu; pij ¼ pj jC max . Hence we have the following corollary. Corollary 3.10. For the problem FmjLEgtp ; prmu; pij ¼ pj ; dj ¼ djLmax , there exists an optimal schedule in which the jobs are ordered according to the SPT rule.
4. Conclusions Scheduling problems with learning effects have recently received growing attention from the scheduling research community. However, most of the research is based on specific learning functions. The aim of the paper is to bring into the scheduling field a general learning effect model where the actual processing time of a job is not only a function of the total normal processing times of the jobs already processed, but also a function of the job’s scheduled position. Particularly, we showed that the single machine makespan and the sum of the kth power of completion times problems remain polynomially solvable under the proposed model. In addition, we showed that the total weighted completion time and the maximum lateness problems are polynomially solvable under certain conditions. Finally, we showed that for some special cases of the m-machine permutation flowshop problems, similar results can also be obtained under the proposed model. 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