Optim LettDOI 10.1007/s11590-012-0522-4

ORIGINAL PAPER

Single machine scheduling with general time-dependentdeterioration, position-dependent learningand past-sequence-dependent setup times

Xue Huang · Gang Li · Yunzhang Huo · Ping Ji

Received: 6 December 2011 / Accepted: 21 June 2012© Springer-Verlag 2012

Abstract The paper deals with single machine scheduling problems with setup timeconsiderations where the actual processing time of a job is not only a non-decreas-ing function of the total normal processing times of the jobs already processed, butalso a non-increasing function of the job’s position in the sequence. The setup timesare proportional to the length of the already processed jobs, i.e., the setup times arepast-sequence-dependent (p-s-d). We consider the following objective functions: themakespan, the total completion time, the sum of the δth (δ ≥ 0) power of job comple-tion times, the total weighted completion time and the maximum lateness. We show thatthe makespan minimization problem, the total completion time minimization problemand the sum of the δth (δ ≥ 0) power of job completion times minimization problemcan be solved by the smallest (normal) processing time first (SPT) rule, respectively.We also show that the total weighted completion time minimization problem and themaximum lateness minimization problem can be solved in polynomial time undercertain conditions.

Keywords Scheduling · Single machine · Learning effect · Deteriorating jobs ·Setup times

X. Huang (B)School of Science, Shenyang Aerospace University, Shenyang, 110136 Liaoning, Chinae-mail: huangxuebj@126.com

G. LiSchool of Management, Xi’an Jiaotong University, The State Key Lab for Manufacturing SystemsEngineering, The Key Lab of the Ministry of Education for Process Control & Efficiency Engineering,Xi’an, 710049, China

Y. Huo · P. JiDepartment of Industrial and Systems Engineering,The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

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1 Introduction

In classical scheduling models, it is reasonable and necessary to consider schedul-ing problems with setup times [1]. There are two types of setup time or setup cost:sequence-independent and sequence-dependent. In the first case, the setup time/costdepends solely on the task to be processed, regardless of its preceding task. While inthe sequence-dependent type, setup time/cost depends on both the task and its pre-ceding task. For recent results and trends in scheduling problems with setup times orcosts, we may refer to the recent review paper of Allahverdi et al. [2]. Koulamas andKyparisis [20] first introduced a scheduling problem with past-sequence-dependent(p-s-d) setup times, i.e., the setup time is dependent on all already scheduled jobs. Theobjectives are the makespan, the total completion time, and the total absolute differ-ences in completion times. They proved that the standard single-machine schedulingwith p-s-d setup times and any of the above objectives can be solvable in polyno-mial times. They also extended their results to nonlinear p-s-d setup times. Kuo andYang [21] considered single-machine scheduling with past-sequence-dependent setuptimes and job-independent (job-dependent) learning effect. They considered the fol-lowing objective function: the makespan, the total completion time, the total absolutedifferences in completion times and the sum of earliness, tardiness, and common due-date penalty. They also proposed the polynomial time algorithms to optimally solvethe above objective functions. Wang [33] studied single-machine scheduling problemswith p-s-d setup times and time-dependent learning effect. They proved that the make-span minimization problem, the total completion time minimization problem and thesum of the quadratic job completion times minimization problem can be solved by theSPT rule, respectively. Wang et al. [34] studied single-machine scheduling problemswith p-s-d setup times and the effects of deterioration and learning. They proved thatthe makespan minimization problem, the total completion time minimization problem,and the sum of the δth power of job completion times minimization problem can besolved in polynomial time, respectively. Hsu et al. [15] considered an unrelated paral-lel machine scheduling problem with setup time and learning effects simultaneously.They showed that the total completion time minimization with the p-s-d setup timecan be solved optimally in the polynomial time.

In classical scheduling problems the processing time of a job is assumed to be aconstant. However, there are many situations that the processing times of jobs may besubject to change due to deterioration and/or learning phenomena [29]. Extensive sur-veys of research related to scheduling deteriorating jobs can be found in Alidaee andWomer [3], Cheng et al. [6], and Gawiejnowicz [11]. An extensive survey of differentscheduling models and problems involving jobs with learning effects can be found inBiskup [4], and Janiak et al. [16].

More recent papers which have considered scheduling jobs with deteriorating jobsand/or learning effects include Wu and Lee [43–45], Cheng et al. [8–10], Lee et al.[28], Wang et al. [40], Wang et al. [41], Sun [31], Janiak et al. [19], Lee and Wu[26,27], Janiak and Rudek [17,18], Lee and Wu [26,27], Lee et al. [28], Yin et al.[51], Lai and Lee [22], Cheng et al. [5], Huang et al. [14], Rudek [30], Cheng et al. [7],Huang et al. [13], Lai et al. [23], Lee [24,25], Wang and Wang [35,36], Wang et al.[37], Wu et al. [46,47], Wu et al. [48], Yang et al. [49], Yang and Yang [50], Wang

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et al. [38], Wang et al. [39], Wang et al. [42], and Yin et al. [52]. Wu and Lee [43] andCheng et al. [8] considered some scheduling problems with learning effect in whichthe actual job processing time is a function of jobs already processed. Wu and Lee WC[44] considered single-machine group scheduling problems with deteriorating setuptimes and job processing times. Cheng et al. [9] considered the following model, i.e.,the actual processing time of job J j if it is scheduled in position r is given by:

p jr = p j

(p0 + ∑r−1

l=1 p[l]p0 + ∑n

l=1 pl

)a

rb, r, j = 1, 2, . . . , n, (1)

where p0 is a given parameter, p j is the normal (basic) processing time of job J j , p[i]denotes the normal (basic) processing time of job scheduled in the i th position in thesequence, a < 0 is a constant deterioration rate, b < 0 is a constant learning rate, and∑0

i=1 p[i] := 0. They proved that several single machine problems and several flowshop problems remain polynomially solvable. Lee et al. [28] consider the followingdeterioration model:

p jr = p j

(1 +

∑r−1l=1 p[l]∑nl=1 pl

)a

, (2)

where a ≥ 0 is the deterioration index. They proved that the makespan minimizationproblem can be solved in polynomial time. Yin et al. [51] considered some schedulingproblems with general position-dependent and time-dependent learning effects. Wanget al. [41] considered the same model of Lee et al. [28], they showed that the totalcompletion time minimization problem can be optimally solved for the case a ≥ 1.Wang et al. [40] considered the following deterioration model:

p jr = p j

(1 +

r−1∑l=1

p[l]

)a

, (3)

where a ≥ 0 is the deterioration index. They showed that the makespan minimiza-tion problem and the total completion time minimization problem for a ≥ 1 can beoptimally solved. Sun [31] considered the following deterioration and learning model:

p jr = p j

(1 +

r−1∑l=1

p[l]

)a

rb, (4)

where a ≥ 1 is the deterioration index and b < 0 is the learning index. They provedthat several single machine problems remain polynomially solvable. Huang et al. [14]considered the following deterioration model:

p jr = p j

(1 +

r−1∑i=1

p[i]

)a

br−1, (5)

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where a and α denote deterioration and learning indices with a > 1 and 0 < b ≤ 1.They proved that several single machine problems remain polynomially solvable.

Note that all of the models (1-5) mentioned above concerning deterioration jobson scheduling problems are based on specific deterioration functions, such as p jr =p j

( p0+xp0+P

)arb, p jr = p j (1 + x

P )a, p jr = p j (1 + x)a, p jr = p j (1 + x)arb and

p jr = p j (1 + x)abr−1, where x = ∑r−1i=1 p[i] and P = ∑n

i=1 pi . Hence Yin et al.[52] proposed a general job position-dependent learning and time-dependent deterio-ration model of the form

p jr = p j f

(r−1∑l=1

αl p[l]

)g(r), (6)

where f and g are two real functions,∑0

l=1 α[l] p[l] = 0, f : [0,+∞) → [1,+∞)

is a differentiable non-decreasing function with f ′ is non-decreasing on [0,+∞) andf (0) = 1, and g : [1,+∞) → (0, 1] is a non-increasing function with g(1) = 1,αl(αl ≥ 0) is a non-increasing sequence of coefficients.

In this paper, we study the single machine scheduling problems with general time-dependent deterioration, position-dependent learning and past-sequence-dependentsetup times. This model is motivated by the ideas of Koulamas and Kyparisis [20]and Yin et al. [52]. The remaining part of this paper is organized as follows. In Sect.2 we formulate the model. In Sect. 3 we consider several single-machine schedulingproblems. The last section presents the conclusions.

2 Problems description

In this section, we first define the notation that is used throughout this paper, followedby the description of the problem.

n: the number of jobsJ = {J1, J2, . . . , Jn}: the job setpi j: the normal processing time for job J j on machine Mi

pi jr: the actual processing time of job J j on machine Mi if it is scheduled in positionr in a sequencepA[ j]: the actual processing time of a job when it is scheduled in the j th position ina sequencef : [0,+∞) → [1,+∞) : a differentiable non-decreasing function with f ′ isnon-decreasing on [0,+∞)

g: [1,+∞) → [0,+∞): a non-increasing function with g(1) = 1s[r ]: the p-s-d setup time of job J[r ]π = [J1, J2, . . . , Jn]: a job sequenceCi j = Ci j (π): the completion time for job J j on machine Mi in π

C j = Cmj: the completion time for job J j in π

Cmax: the makespan of a given permutation, i.e., Cmax = max{C j | j = 1, 2, . . . , n}∑C j: the total completion time of a given permutation

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Single machine scheduling with general time-dependent deterioration

∑Cθ

j : the sum of the θ th power of the completion times (θ ≥ 0)

∑w j C j: the weighted sum of completion times

Lmax = max{C j − d j | j = 1, 2, . . . , n}: the maximum lateness

The single machine scheduling problem is formulated as follows. There are given aset J = {J1, J2, . . . , Jn} of n independent jobs. All jobs are available for processing attime 0. The machine can handle one job at a time and preemption is not allowed. Asso-ciated with each job J j ( j = 1, 2, ..., n) there is a normal processing time p j and p[r ]be the normal processing time of a job if scheduled in the r th position in a sequence. Letp jr be the actual processing time of job J j if it is scheduled in position r in a sequence.As in Yin et al. [52], we consider the model (6), i.e., p jr = p j f (

∑r−1l=1 αl p[l])g(r). In

addition, let pA[ j] be the actual processing time of a job when it is scheduled in the j thposition in a sequence. As in Koulamas and Kyparisis [20] and Kuo and Yang [21], weassume that the p-s-d setup time of job J[r ] if it is scheduled in position r is given by :

s[1] = 0 and s[r ] = dr−1∑i=1

pA[i] (7)

where d ≥ 0 is a normalizing constant. For convenience, we denote the model givenin (6) by ELDgpt [52], and the model given in (7) by spsd [20].

For a given schedule π = [J1, J2, . . . , Jn], C j = C j (π) represents the comple-tion time of job J j and f (C) = f (C1, C2, . . . , Cn) is a regular measure of perfor-mance. Let Cmax = max{C j | j = 1, 2, . . . , n}, ∑

C j ,∑

Cθj ,

∑w j C j and Lmax =

max{C j − d j | j = 1, 2, . . . , n} represent the makespan, the total completion time, thesum of the θ th power of the completion times, the weighted sum of completion timesand the maximum lateness of a given permutation, respectively. In the remaining partof the paper, all the problems considered will be denoted using the three-field notationscheme α|β|γ introduced by Graham et al. [12].

3 Several single machine scheduling problems

First, we give a lemma, which is useful for the following theorems.

Lemma 1 If f : [0,+∞) → [1,+∞) is a differentiable non-decreasing functionwith f ′ is non-decreasing on [0,+∞), then (1 + d)(λ − 1) f (x) + μ f (x + λt) −λμ f (x + t)) ≥ 0 for λ ≥ 1, 0 < μ ≤ 1, t ≥ 0 and x ≥ 0.

Proof Let h(t) = (1+d)(λ−1) f (x)+μ f (x +λt)−λμ f (x + t)). By taking the firstderivative of h(t) with respect to t , we have h′(t) = μλ[ f ′(x + λt) − f ′(x + t)] ≥ 0for λ ≥ 1, 0 < μ ≤ 1, t ≥ 0 and x ≥ 0. Hence, h(t) is increasing on t ≥ 0, and wehave h(t) ≥ h(0) = (λ − 1)(1 + d − μ) f (x) ≥ 0. This completes the proof. ��Theorem 1 For the problem 1|ELDgpt, spsd |Cmax, an optimal schedule can beobtained by sequencing the jobs in non-decreasing order of p j (the SPT rule).

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Proof The proof follows directly from the pairwise interchange analysis. Let π and π ′be two job schedules where the difference between π and π ′ is a pairwise interchangeof two adjacent jobs J j and Jk , that is, π = [S1, J j , Jk, S2], π ′ = [S1, Jk, J j , S2],where S1 and S2 are partial sequences. Furthermore, we assume that there are r − 1jobs in S1. Thus, J j and Jk are the r th and the (r + 1)th jobs, respectively, in π .Likewise, Jk and J j are scheduled in the r th and the (r + 1)th positions in π ′. Inaddition, let A denote the completion time of the last job in π1. Suppose p j ≤ pk . Toshow π dominates π ′, it suffices to show that the (r + 1)th jobs in π and π ′ satisfy thecondition that Ck(π) ≤ C j (π

′). Under π , the completion times of jobs J j and Jk are

C j (π) = A + dr−1∑i=1

pA[i] + p j f

(r−1∑l=1

αl p[l]

)g(r). (8)

and

Ck(π) = A + dr−1∑i=1

pA[i] + p j f

(r−1∑l=1

αl p[l]

)g(r)

+ d

(r−1∑i=1

pA[i] + p j f

(r−1∑l=1

αl p[l]

)g(r)

)

+pk f

(r−1∑l=1

αl p[l] + αr p j

)g(r + 1). (9)

Whereas under S′, they are

Ck(π′) = A + d

r−1∑i=1

pA[i] + pk f

(r−1∑l=1

αl p[l]

)g(r). (10)

and

C j (π′) = A + d

r−1∑i=1

pA[i] + pk f

(r−1∑l=1

αl p[l]

)g(r)

+ d

(r−1∑i=1

pA[i] + pk f

(r−1∑l=1

αl p[l]

)g(r)

)

+p j f

(r−1∑l=1

αl p[l] + αr pk

)g(r + 1). (11)

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Single machine scheduling with general time-dependent deterioration

Taking the difference between (9) and (11), we obtain

C j (π′) − Ck(π) = (1 + d)(pk − p j ) f

(r−1∑l=1

αl p[l]

)g(r)

+p j f

(r−1∑l=1

αl p[l] + αr pk

)g(r + 1)

−pk f

(r−1∑l=1

αl p[l] + αr p j

)g(r + 1). (12)

Let t = αr p j , λ = pkp j

, x = ∑r−1l=1 αl p[l] and μ = g(r + 1)/g(r). Then inequality

(12) can be rewritten as

C j (π′) − Ck(π) = p j g(r)[(1 + d)(λ − 1) f (x) + μ f (x + λt) − λμ f (x + t))].

From Lemma 1, we have C j (π′) − Ck(π) ≥ 0. This completes the proof. ��

Theorem 2 For the problem 1|ELDgpt, spsd | ∑ C j , an optimal schedule can beobtained by sequencing the jobs in non-decreasing order of p j (i.e., the SPT order).

Proof We adopt the same notations as in the proof of Theorem 1. Let π and π ′ betwo job schedules where the difference between π and π ′ is a pairwise interchange oftwo adjacent jobs J j and Jk , i.e., π = [S1, J j , Jk, S2], π ′ = [S1, Jk, J j , S2], whereS1 and S2 are partial sequences. Furthermore, we assume that there are r − 1 jobs inS1. Thus, J j and Jk are the r th and the (r + 1)th jobs, respectively, in π and p j ≤ pk .Likewise, Jk and J j are scheduled in the r th and the (r + 1)th positions in π ′. Toshow π dominates π ′, it suffices to show that the (r + 1)th jobs in π and π ′ satisfythe condition that Ck(π) ≤ C j (π

′) and C j (π) + Ck(π) ≤ Ck(π′) + C j (π

′).From Theorem 1, we have C j (π

′) ≥ Ck(π), and from p j ≤ pk , (8) and (10), wehave Ck(π

′) ≥ C j (π), hence

Ck(π′) + C j (π

′) ≥ C j (π) + Ck(π).

Repeating this argument will lead to the optimality of the SPT-sequence for the prob-lem 1|ELDgpt, spsd | ∑ C j . This completes the proof. ��

Townsend [32] considered single-machine scheduling with the quadratic objective.He showed that the problem 1||∑ C2

j can be solved optimally by the SPT rule. By asimilar proof as that for Theorem 2, we can show that the solution of Townsend’s stillholds for the problems 1|ELDgpt, spsd | ∑ Cθ

j for any θ ≥ 0.

Theorem 3 For the problem 1|ELDgpt, spsd | ∑ Cθj , an optimal schedule can be

obtained by sequencing the jobs in non-decreasing order of p j (i.e., the SPT order).

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Proof We adopt the same notations as in the proof of Theorem 1. It is similar to theproof of Theorem 2, except that C j (π

′) ≥ Ck(π), Ck(π′) ≥ C j (π). Hence,

Cθk (π ′) + Cθ

j (π′) ≥ Cθ

j (π) + Cθk (π).

This completes the proof. ��Theorem 4 For the problem 1|ELDgpt, spsd | ∑w j C j , if the jobs have agreeableweights, i.e., p j ≤ pk implies w j ≥ wk for all the jobs J j and Jk, then an optimalschedule can be obtained by sequencing the jobs in non-decreasing order of p j/w j

(i.e., the WSPT order).

Proof Consider an optimal schedule π that does not follow the WSPT order. In thisschedule there must be at least two adjacent jobs, say job J j followed by job Jk , suchthat p j/w j > pk/wk , which implies p j ≥ pk and w j ≤ wk . Assume that job J j isscheduled in position r . Perform an adjacent pair-wise interchange of jobs J j and Jk .Whereas under the original schedule π job J j is scheduled in position r and job Jk isscheduled in position r + 1, under the new schedule job Jk is scheduled in position rand job J j is scheduled in position r + 1. In addition, let t be the starting time of jobJ j in π . All the other jobs remain in their original positions. Call the new schedule π ′.The completion times of the jobs processed before jobs J j and Jk are not affected bythe interchange. Furthermore, the completion times of the jobs processed after jobsJ j and Jk will not be increased by the interchange since p j ≥ pk .

From Eqs. (8), (9), (10) and (11), we have

w j C j (π) + wkCk(π) − wkCk(π′) − w j C j (π

′)

=(w j +wk)(p j − pk) f

(r−1∑l=1

αl p[l]

)g(r)+wk pk f

(r−1∑l=1

αl p[l]+αr p j

)g(r +1)

−w j p j f

(r−1∑l=1

αl p[l] + αr pk

)g(r + 1)

+ d(wk − w j )

r−1∑i=1

pA[i] + d(wk p j − w j pk) f

(r−1∑l=1

αl p[l]

)g(r)

= (w j + wk)pk g(r)

[(p j

pk− 1

)f

(r−1∑l=1

αl p[l]

)

+ wk

w j + wk

g(r + 1)

g(r)f

(r−1∑l=1

αl p[l] + αr p j

)

− w j

w j + wk

p j

pk

g(r + 1)

g(r)f

(r−1∑l=1

αl p[l] + αr pk

)]

+ d(wk − w j )

r−1∑i=1

pA[i] + d(wk p j − w j pk) f

(r−1∑l=1

αl p[l]

)g(r) (13)

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Single machine scheduling with general time-dependent deterioration

Let δ1 = w jw j +wk

· g(r+1)g(r)

, δ2 = wkw j +wk

· g(r+1)g(r)

, t = αr pk, x = ∑r−1l=1 αl p[l] and

λ = p jpk

, we have

(p j

pk− 1

)f

(r−1∑l=1

αl p[l]

)+ wk

w j + wk

g(r + 1)

g(r)f

(r−1∑l=1

αl p[l] + αr p j

)

− w j

w j + wk

p j

pk

g(r + 1)

g(r)f

(r−1∑l=1

αl p[l] + αr pk

)

= (λ − 1) f (x) + δ2 f (x + λt) − δ1λ f (x + t)

≥ (λ − 1) f (x) + δ1 f (x + λt) − δ1λ f (x + t)

≥ 0 (Lemma 1).

From wk p j > w j pk and w j ≤ wk , we have w j C j (π) + wkCk(π) ≥ wkCk(π′) +

w j C j (π′). It follows that the weighted sum of completion times under π ′ is less than

or equal to that under π . Repeating this argument will lead to the result of the theorem.��

Using the similar method of Theorem 4, the following theorems can be easilyobtained.

Theorem 5 For the problem 1|ELDgpt, spsd , p j = p| ∑w j C j , an optimal schedulecan be obtained by sequencing the jobs in non-increasing order of w j .

Theorem 6 For the problem 1|ELDgpt, spsd , w j = w| ∑w j C j , an optimal schedulecan be obtained by sequencing the jobs in non-decreasing order of p j (i.e., the SPTorder).

Theorem 7 For the problem 1|ELDgpt, spsd , w j p j = θ | ∑ w j C j , an optimal sched-ule can be obtained by sequencing the jobs in non-decreasing order of p j (i.e., theSPT order).

Theorem 8 For the problem 1|ELDgpt, spsd |Lmax, if the jobs have agreeable condi-tions, i.e., p j < pk implies d j ≤ dk for all the jobs J j and Jk, then an optimal schedulecan be obtained by sequencing the jobs in non-decreasing order of d j (i.e., the EDDorder).

Proof Consider an optimal schedule π that does not follow the EDD order. In thisschedule there must be at least two adjacent jobs, say J j and Jk in the r th and (r +1)thpositions of π , respectively, such that d j > dk , which implies p j ≥ pk . Schedule π ′is obtained from schedule π by interchanging jobs in the r th and in the (r + 1)thpositions of π . From the proof of Theorem 1, under π , the lateness of the jobs are

L j (π) = C j (π) − d j ,

Lk(π) = Ck(π) − dk,

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X. Huang et al.

whereas under π ′, they are

Lk(π′) = Ck(π

′) − dk,

L j (π′) = C j (π

′) − d j .

Since d j > dk and p j ≥ pk , we have C j (π′) ≤ Ck(π) and Ck(π

′) ≤ C j (π)

(Theorem 1). Hence, it is easy to verify that

max{L j (π′), Lk(π

′)} < max{L j (π), Lk(π)}.

Hence, interchanging the positions of jobs J j and Jk will decrease the value of Lmax.This is a contradiction. ��

Using the similar method of Theorem 8, the following theorems can be easilyobtained.

Theorem 9 For the problem 1|ELDgpt, spsd , p j = p|Lmax, an optimal schedule canbe obtained by sequencing the jobs in non-decreasing order of d j (i.e., the EDD order).

Theorem 10 For the problem 1|ELDgpt, spsd , d j = d|Lmax, an optimal schedule canbe obtained by sequencing the jobs in non-decreasing order of p j (i.e., the SPT order).

If d j = θp j , the jobs have agreeable conditions, i.e., pi ≤ p j implies di ≤ d j forall the jobs Ji and J j . Hence, the following corollary can be easily obtained.

Corollary 1 For the problem 1|ELDgpt, spsd , d j = θp j |Lmax, an optimal schedulecan be obtained by sequencing the jobs in non-decreasing order of d j (i.e., the EDDorder).

4 Conclusions

In this paper we considered some single machine scheduling problems with generaltime-dependent deterioration, position-dependent learning and past-sequence-depen-dent setup times. We proved that the makespan minimization problem, the total com-pletion time minimization problem, and the sum of the θ th power of job completiontimes minimization problem can be solved by the SPT rule. We also proved that somespecial cases of the total weighted completion time minimization problem and themaximum lateness minimization problem can be solved in polynomial time. Futureresearch may focus on considering other objective functions, or proposing more gen-eral and practical models.

Acknowledgments We are grateful to the editor and two anonymous referees for their helpful commentson earlier versions of this paper. The work described in this paper was fully supported by a grant fromthe Research Grant Council of the Hong Kong Special Administrative Region, China (Project No. PolyU517011).

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Single machine scheduling with general time-dependent deterioration

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