Single-machine scheduling with deteriorating jobs and past-sequence-dependent setup times

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Single-machine scheduling with deteriorating jobs

and past-sequence-dependent setup times

T.C.E. Cheng

a

, Wen-Chiung Lee

b,⇑

, Chin-Chia Wu

b a

Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

b

Department of Statistics, Feng Chia University, Taichung, Taiwan

a r t i c l e

i n f o

Article history:

Received 10 November 2009

Received in revised form 16 September 2010 Accepted 4 October 2010

Available online 13 October 2010 Keywords:

Deteriorating jobs Single-machine

Past-sequence-dependent setup time

a b s t r a c t

In many realistic scheduling settings a job processed later consumes more time than the same job processed earlier – this is known asscheduling with deteriorating jobs. Most research on scheduling with deteriorating jobs assumes that the actual processing time of a job is an increasing function of its starting time. Thus a job processed late may incur an excessively long processing time. On the other hand, setup times occur in manufactur-ing situations where jobs are processed in batches whereby each batch incurs a setup time. This paper considers scheduling with deteriorating jobs in which the actual processing time of a job is a function of the logarithm of the total processing time of the jobs processed before it (to avoid the unrealistic situation where the jobs scheduled late will incur exces-sively long processing times) and the setup times are proportional to the actual processing times of the already scheduled jobs. Under the proposed model, we provide optimal solu-tions for some single-machine problems.

1. Introduction

In classical scheduling theory, the processing times of jobs are assumed to be known and fixed; however, there are many situations in which a job that is processed later consumes more time than the same job processed earlier, e.g., steel produc-tion, fire fighting, and single-server cycle-queues[1–3]. In all of these cases, executing a task may take more time as time passes. Scheduling problems with deteriorating jobs were first introduced independently by Gupta and Gupta [1] and Browne and Yechiali [3]. Since then, the topic has received increasing attention of the scheduling research community and researchers have proposed various scheduling models in which the processing time of a job is assumed to be an increas-ing function of its startincreas-ing time – this is known asscheduling with deteriorating jobs. Comprehensive reviews of this stream of scheduling research can be found in Alidaee and Womer[4]and Cheng et al.[5].

Recently, Gawiejnowicz et al.[6]consider two problems of single-machine bi-criterion scheduling of a set of deteriorating jobs. The first problem is to find a Pareto optimal schedule with respect to the total completion time and the maximum com-pletion time. The second problem is to minimize a convex combination of these two criteria. Ji et al.[7]consider a single-ma-chine problem in which the processing time of a job is a simple linear increasing function of its starting time and the masingle-ma-chine is subject to an availability constraint. Gawiejnowicz et al.[8]consider the single-machine problem with linearly deteriorat-ing jobs to minimize the total completion time. Wu and Lee[9]study the two-machine flowshop problem with linear job dete-rioration to minimize the mean flow time. Shiau et al.[10]consider a simple linear job deterioration model in the two-machine flowshop to minimize the mean flow time. Kang and Ng[11]prove the NP-hardness of the problem of scheduling

⇑Corresponding author.

E-mail address:[email protected](W.-C. Lee).

Contents lists available atScienceDirect

Applied Mathematical Modelling

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a p m

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nlinearly deteriorating jobs onmidentical parallel machines to minimize the makespan. Gawiejnowicz[12]consider two problems of scheduling a set of independent, non-preemptive and proportionally deteriorating jobs on a single machine. Wu et al.[13]investigate a single-machine problem in which the job processing times are starting time dependent to min-imize the total weighted completion time. Raut et al.[14]consider the problem of scheduling on a single machine with time deteriorating job values and capacity constraints. Lee et al.[15]investigate single-machine scheduling with deteriorating jobs and job release times to minimize the makespan. Lee et al.[16]provide a dominance rule and a lower bound to construct a branch-and-bound algorithm for them-machine permutation ﬂow shop problem to minimize the total completion time. Re-cently, Biskup and Herrmann[17]observe that the sum of the processing times of the jobs processed before a job contributes to the actual processing time of the job and they cite equipment wear-out (e.g., a drill) as a real-life example of their obser-vation. Wang and Guo[18]consider a single-machine scheduling problem with the effects of learning and deterioration. The goal is to determine an optimal combination of the due-date and schedule so as to minimize the sum of earliness, tardiness, and due-date costs. Ng et al.[19]consider a two-machine ﬂow shop scheduling problem with linearly deteriorating jobs to minimize the total completion time.

The above works on scheduling with deteriorating jobs neglect the setup cost or setup times. However, scheduling with setup times or setup cost plays a crucial role in today’s manufacturing and service environments where reliable products/ services are to be delivered on time. Scheduling problems involving setup times can be divided into two classes, namely se-quence-independent and sequence-dependent setup times. Setup is sequence-dependent if its duration depends on both the current and the immediately preceding job, and is sequence-independent if its duration depends only on the current job to be processed. Sequence-dependent setup times are usually found in the situation where the facility is a multi-purpose ma-chine. The case of sequence-dependent setups can be found in many other industrial settings, which include stamping oper-ations in plastic manufacturing, die changing in metal processing shops, and roll slitting in the paper industry. Surveys of scheduling research involving setup times were given by Cheng et al.[20]and Allahverdi et al.[21,22]. Recently, Koulamas and Kyparisis[23]point out that in high-tech manufacturing jobs are commonly processed in batches, e.g., a batch of jobs may consist of a group of electronic components mounted together on an integrated circuit (IC) board, whereby each batch incurs a setup time. Wu et al.[24]consider two single-machine scheduling problems in the context of group technology where the job processing times and setup times are simple linear functions of their starting times. Wu and Lee[25]study a single-machine problem with the assumption that the group setup times and job processing times are both increasing functions of their starting times to minimize the makespan. Lee and Wu[26]investigate a multi-machine scheduling prob-lem in which the job processing times are increasing functions of their starting times and the machines are not always avail-able. Allahverdi and Soroush[27]claim that setup activities due to changeovers represent costly disruptions to production/ service processes. Therefore, setup reduction is an important feature of the continuous improvement programme of any manufacturing/service organization. It is especially critical if an organization seeks to make timely responses to market changes through shortened lead times and production in smaller lot sizes. Every scheduler should understand the principles of setup reduction and be able to recognize its potential beneﬁts. Koulamas and Kyparisis[23]introduce

past-sequence-dependent (p-s-d)setup times to scheduling problems whereby the setup times are proportional to the actual processing

times of the already scheduled jobs. They show that several single-machine scheduling problems to minimize some comple-tion time-based objective measures remain polynomially solvable. Biskup and Herrmann[17]extended their analysis to problems with due dates. They demonstrate that some problems are polynomially solvable. Eren[28]provide an example in which jobs are processed by automatic machines so the job processing times are the same regardless of their processing order. However, human factor becomes inﬂuential when setup times and removal times are taken into consideration. Setup times will increase when workers perform many setup operations over time due to fatigue. Thus, in this paper we study scheduling with deteriorating jobs that incorporates p-s-d setup times.

The remainder of this paper is organized as follows. We introduce the problem formulation and notation in the next sec-tion. In Section3we derive the optimal solutions for some single-machine problems. We conclude the paper in the ﬁnal section.

2. Problem formulation

Before presenting the main results, we introduce some notation that will be used throughout the paper.

n number of jobs

pj normal processing time of jobj

dj due date of jobj

pA

jr actual processing time of jobjscheduled in therth position

p[r] normal processing time of a job scheduled in therth position

a deterioration index, wherea> 1

s[r] p-s-d setup time of jobjscheduled in therth position

Cj completion time of jobj

Lj lateness of jobj, i.e.,Lj=Cjdj

Tj tardiness of jobj, i.e.,Tj= max{Lj, 0}

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P

Cj total completion time of all the jobs

P

C2j sum of the quadratic job completion times of all the jobs

Lmax maximum lateness of all the jobs, i.e.,Lmax= {Lj|j= 1, 2,. . .,n}

P

Tj total tardiness of all the jobs

M1,M2, M3, M four positive numbers, where M1¼1þPnj¼1logpj; M2¼log maxpi–pjfpi=pjg

n o

; M3¼max1inflogpig, and

M= 1 +M1+ logM2+M3

The formulation of the proposed problem is as follows: There arenjobs ready to be processed on a single machine. Each jobjhas a normal processing timepjand a due datedj. We model the deterioration effect based on the observation of Biskup

and Herrmann[17]that the processing times of the jobs processed before a job contribute to the actual processing time of that job. Job deterioration may be caused by various factors such as equipment wear-out and machine re-alignment in pre-cision manufacturing. However, a job processed late may incur an excessively long processing time if there are many jobs processed before it. To avoid this unrealistic situation, we assume that the actual processing time of a job is a function of the sum of the logarithm of the normal processing times of the jobs processed before it. Thus, if a jobjis scheduled in therth position in a sequence, then its actual processing time is

pA j½r¼pj 1þ Xr1 l¼1 logp½l !a ;

wherep[l]is the normal processing time of the job scheduled in thelth position in the sequence anda>Mis the deterioration

index.

We also take setup times into consideration in the scheduling model by adopting the notion of Koulamas and Kyparisis [23]and Wang[29]that setup times are past-sequence-dependent. Koulamas and Kyparisis[23]motivate the assumption of p-s-d setup times by the manufacturing of integrated circuit (IC) boards. Speciﬁcally, an IC board consists of a number of electronic components mounted on it and the processing of any electronic component will have an adverse effect on the ‘‘readiness” of other components on the board due to the passage of electricity through the board. Consequently, each com-ponent prior to processing requires a setup operation to restore it to ‘‘full-readiness” status and the setup time depends on the component’s degree of ‘‘un-readiness”, which is proportional to the actual processing times of the already processed components. Thus, the p-s-d setup time of jobjif it is scheduled in therth position in a sequence is given as follows:

S½1¼0 ands½r¼

c

Xr1

l¼1

pA ½l;

where

c

is a normalizing constant with 0 <

c

< 1 andpA

½ldenotes the actual processing time of a job if it is scheduled in thelth position.

For notational convenience, we denote all the problems under consideration using the three-ﬁeld notation scheme

a

|b|d

for scheduling problems introduced by Graham et al.[30].

3. Single-machine problems

Before presenting the main results, we ﬁrst present two lemmas, which will be used in the proofs of the theorems in the sequel.

Lemma 1.

c

+ 1 +ac0(1+c0x)a1(1 +c0x)aP0for0 <

c

< 1, 1/M1<c0< 1,a>M,and06x6M₃.

Proof. LetF(x) =

c

+ 1 +ac0(1 +c0x)a1(1 +c0x)a. Taking the ﬁrst derivative ofF(x) with respect tox, we have F0_{ðxÞ ¼}_aða 1Þc2 0ð1þc0xÞ a2 ac0ð1þc0xÞ a1 ¼ac0ð1þc0xÞ a2 ða1Þc0 ð1þc0xÞ ½ >0

sincea>Mand 1 + 1/c0+x< 1 +M1+M3. This implies thatF(x) is an increasing function for 06x6M3. SinceF(x)PF(0) =

c

+ac0> 0 for 0 <

c

< 1,a>Mand 1/M1<c0< 1,F(x)P0 for 0 <

c

< 1, 1/M1<c0< 1,a>M, and 06x6M3. This completes

the proof. h

Lemma 2. (

c

+1)(h1) +(1 +c0logh+c0x)ah(1 +c0x)aP0for16h6M2, 0 <

c

< 1, 1/M1<c0< 1,a>M,and06x6M3. Proof. LetG(h) = (

c

+ 1)(h1) + (1 +c0logh+c0x)ah(1 +c0x)a. Taking the ﬁrst and second derivatives ofG(h) with respect

toh, we have G0 ðhÞ ¼ ð

c

þ1Þ þac0ð1þc0loghþc0xÞ a1 =h ð1þc0xÞ a and G00

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Sincea>Mand 1 + 1/c0+ logh+x< 1 +M1+ logM2+M3,G00ðhÞ>0. ThereforeG0(h) is an increasing function for 16h6M2.

FromLemma 1, we have

G0

ð1Þ ¼ ð

c

þ1Þ þac0ð1þc0xÞa1 ð1þc0xÞaP0:

SinceG0₍_h_{) is an increasing function for 1}6_h6_M₂_,_G0₍_h₎_P_G0₍₁₎_P_{0. Therefore,}_G₍_h_{) is an increasing function for 1}6_h6_M₂_.

SinceG(1) = 0,G(h)P0 for 16h6M₂, 0 <

c

< 1, 1/M₁<c₀< 1,a>M, and 06x6M₃. This completes the proof. h Theorem 1. For the1jpj½r¼pj 1þ

Pr1

l¼1logp½l

a

; spsdjCmaxproblem, the optimal schedule is obtained by sequencing jobs in the

shortest processing time (SPT) order.

Proof.Supposepi6pj. LetSandS0be two job schedules where the difference betweenSandS0is a pairwise interchange of

two adjacent jobsiandj, i.e.,S= (

p

i j

p

0_{) and}_S0_{= (}

p

_{i j}

p

0_{), where}

p

_and

p

0_{denote partial sequences. Furthermore, we assume} that there arer1 jobs in

p

. Thus, jobsiandjare therth and (r+ 1)th jobs inS, respectively, whereas jobsjandiare sched-uled in therth and (r+ 1)th positions inS0_{, respectively. In addition, let}_A_{denote the completion time of the last job in}

p

_{. To} show thatSdominatesS0_{, it sufﬁces to show that the (}_r_{+ 1)th jobs in}_S_and_S0_{satisfy the condition}_C

j(S)6Ci(S0). By deﬁnition,

the actual processing times of jobjinSand jobiinS0_{are given by}

CjðSÞ ¼Aþpi 1þX r1 l¼1 logp½l !a þ

c

X r1 l¼1 pA ½lþpj 1þ Xr1 l¼1 logp½lþlogpi !a þ

c

X r1 l¼1 pA ½lþpi 1þ Xr1 k¼1 logp½k !a! ð1Þ and CiðS0Þ ¼Aþpj 1þX r1 l¼1 logp½l !a þ

c

X r1 l¼1 pA ½lþpi 1þ Xr1 l¼1 logp½lþlogpj !a þ

c

X r1 l¼1 pA ½lþpj 1þ Xr1 k¼1 logp½k !a! : ð2Þ

Taking the difference between(1) and (2), we obtain

CiðS0 Þ CjðSÞ ¼ ð

c

þ1ÞðpjpiÞ 1þ Xr1 l¼1 logp½l !a þpi 1þ Xr1 l¼1 logp½lþlogpj !a pj 1þ Xr1 l¼1 logp½lþlogpi !a : ð3Þ Substitutingh=pj/pi,c0¼1= 1þPrl¼11logp½l

, andx= logpiinto(3)and simplifying, we obtain

CiðS0_Þ_Cjð_S_{Þ ¼}_pi ₁_þX r1 l¼1 logp½l !a ð

c

þ1Þðh1Þ þ ð1þc0loghþc0xÞahð1þc0xÞa : ð4Þ

FromLemma 2and since 16_h6_M₂_{, 0 <}

c

_{< 1, 1/}_M₁_<_c₀_{< 1,}_a_>_M_{, and 0}6_x6_M₃_,_C_i₍_S0₎_C_j₍_S₎_P_{0. Thus,}_S_dominates_S0_.

Therefore, repeating this interchange argument for all the jobs not sequenced in the SPT order will yield the desired result. h

Theorem 2. For the1jp_j_½_r¼p_j 1þPrl¼11logp½l

a

; spsdjPCjproblem, the optimal schedule is obtained by sequencing jobs in the

SPT order.

Proof.Suppose thatpi6p_j. LetSandS0be two job schedules where the difference betweenSandS0is a pairwise interchange

of two adjacent jobsiandj, i.e.,S= (

p

i j

p

0_{) and}_S0_{= (}

p

_{i j}

p

0_{), where}

p

_and

p

0_{denote partial sequences. Furthermore, we} assume that there arer1 jobs in

p

. Thus, jobsiandjare therth and (r+ 1)th jobs inS, respectively, whereas jobsjand iare scheduled in therth and (r+ 1)th positions inS0_{, respectively. In addition, let}_A_{denote the completion time of the last} job in

p

. To show thatSdominateS0_{, it sufﬁces to show that the (}_r_{+ 1)th jobs in}_S_and_S0_{satisfy the conditions}_C

j(S) <Ci(S0) and

Ci(S) +Cj(S)6Cj(S0) +Ci(S0). By deﬁnition, the actual processing times of jobsiandjinSare given by

CiðSÞ ¼Aþpi 1þX r1 l¼1 logp½l !a þ

c

X r1 l¼1 pA ½l; ð5Þ CjðSÞ ¼Aþpi 1þX r1 l¼1 logp½l !a þ

c

X r1 l¼1 pA ½lþpi 1þ Xr1 k¼1 logp½k !a! : ð6Þ

Similarly, the actual processing times of jobsiandjinS0_are

CjðS0 Þ ¼Aþpj 1þ Xr1 l¼1 logp½l !a þ

c

X r1 l¼1 pA ½l; ð7Þ

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and CiðS0 Þ ¼Aþpj 1þ Xr1 l¼1 logp½l !a þ

c

X r1 l¼1 pA ½lþpj 1þ Xr1 k¼1 logp½k !a! : ð8Þ

SincepjpiP0,Cj(S) <Ci(S0) from Theorem 1. Therefore, we only need to show thatCi(S) +Cj(S)6Cj(S0) +Ci(S0). From(5)to

(8), we have

fCjðS0Þ þCiðS0Þg fCiðSÞ þCjðSÞg ¼ ðpjpiÞ 1þX r1 l¼1 logp½l !a þ ð

c

þ1ÞðpjpiÞ 1þX r1 l¼1 logp½l !a þpi 1þ Xr1 l¼1 logp½lþlogpj !a pj 1þ Xr1 l¼1 logp½lþlogpi !a : ð9Þ

SincepjpiP0 and 1þPlr¼11logp½l

a

>0, the ﬁrst term of(9)is non-negative. From the proof ofTheorem 1, we know that the sum of the last three terms of(9)is non-negative, too. Hence,Ci(S) +Cj(S)6Cj(S0) +Ci(S0). Thus, repeating this interchange

argument for all the jobs not sequenced in the SPT order yields the desired result. h Theorem 3. For the1jpj½r¼pj 1þ

Pr1

l¼1logp½l

a

; spsdjPC2j problem, the optimal schedule is obtained by sequencing jobs in the

SPT order.

Proof. It is similar to that ofTheorem 2and is omitted.In the following we show that the earliest due date (EDD) rule pro-vides the optimal solution for the total tardiness problem if the job processing times and due dates are agreeable, i.e.,di6dj

impliespi6pjfor all jobsiandj. Note that the EDD rule is also the SPT rule if the job processing times and due dates are

agreeable. h

Theorem 4. For the1jpj½r¼pj 1þ

Pr1

l¼1logp½l

a

; spsdjPTiproblem, the optimal schedule is obtained by sequencing jobs in

non-decreasing order ofdi,i.e., the EDD order, if the job processing times and due dates are agreeable.

Proof. Suppose thatdi6dj, which impliespi6pj. The total tardiness of the ﬁrstr1 jobs are the same since they are

pro-cessed in the same order. Since the makespan is minimized by the SPT rule (Theorem 1), the total tardiness of partial sequence

p

0_in_S_{will not be greater than that of partial sequence}

p

0_in_S0_{. Thus, to prove that the total tardiness of}_S_{is less} than or equal to that ofS0_{, it sufﬁces to show that}_T

i(S) +Tj(S)6Tj(S0) +Ti(S0) .From(5)–(8), we derive that the tardiness of jobsi

andjinSare TiðSÞ ¼max Aþpi 1þ Xr1 l¼1 logp½l !a þ

c

X r1 l¼1 pA ½ldi;0 ( ) and TjðSÞ ¼max Aþpi 1þX r1 l¼1 logp½l !a þ

c

X r1 l¼1 pA ½lþpi 1þ Xr1 k¼1 logp½k !a! dj;0 ( ) :

Similarly, the tardiness of jobsiandjinS0_are

TjðS0Þ ¼max Aþpj 1þX r1 l¼1 logp½l !a þ

c

X r1 l¼1 pA ½ldj;0 ( ) and

TiðS0_{Þ ¼}_max _Aþpj ₁_þX r1 l¼1 logp½l !a þ

c

X r1 l¼1 pA ½lþpj 1þ Xr1 k¼1 logp½k !a! di;0 ( ) :

To compare the total tardiness of jobs i and j in S and in S0_{, we consider two cases. In the ﬁrst case where}

Aþpj 1þ

Pr1

l¼1logp½l

a

þ

c

Pr1

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TiðSÞ þTjðSÞ ¼ max Aþpi 1þ Xr1 l¼1 logp½l !a þ

c

X r1 l¼1 pA ½ldi;0 ( ) þmax Aþpi 1þ Xr1 l¼1 logp½l !a ( þ

c

X r1 l¼1 pA ½lþpi 1þ Xr1 k¼1 logp½k !a! dj;0 ) and TjðS0 Þ þTiðS0 Þ ¼ max Aþpj 1þ Xr1 l¼1 logp½l !a þ

c

X r1 l¼1 pA ½lþpi 1þ Xr1 l¼1 logp½lþlogpj !a ( þ

c

X r1 l¼1 pA ½lþpj 1þ Xr1 k¼1 logp½k !a! di;0 ) :

Suppose that neitherTi(S) norTj(S) is zero. Note that this is the most restrictive case since it comprises the case where

either one or bothTi(S) andTj(S) are zero. FromTheorem 1and the fact thatdi6dj, we have

TjðS0 Þ þTiðS0 Þ fTiðSÞ þTjðSÞg ¼ ð

c

þ1ÞðpjpiÞ 1þ Xr1 l¼1 logp½l !a þpi 1þ Xr1 l¼1 logp½lþlogpj !a pj 1þ Xr1 l¼1 logp½lþlogpi !a þdjpi 1þ Xr1 l¼1 logp½l !a

c

X r1 l¼1 logp½lAP0:

Thus, {Tj(S0) +Ti(S0)}{Ti(S) +Tj(S)}P0. In the second case whereAþpj 1þ

Pr1

l¼1logp½l

a

þ

c

Pr1

l¼1pA½l>dj, the total tardiness

of jobsiandjinSand inS0_are

TiðSÞ þTjðSÞ ¼ Aþpi 1þX r1 l¼1 logp½l !a þ

c

X r1 l¼1 pA ½ldi;0 ( ) þmax Aþpi 1þX r1 l¼1 logp½l !a ( þ

c

X r1 l¼1 pA ½lþpi 1þ Xr1 k¼1 logp½k !a! dj;0 ) and TjðS0_{Þ þ}_TiðS0_{Þ ¼}₂_A_þ₂_pj ₁_þX r1 l¼1 logp½l !a þpi 1þX r1 l¼1 logp½lþlogpj !a didjþ

c

pj 1þX r1 l¼1 logp½l !a þ2cX r1 l¼1 logpA ½l:

Suppose that neitherTi(S) norTj(S) is zero. FromTheorem 1andpi6p_j, we have

fTjðS0 Þ þTiðS0_{Þg fTiðSÞ þ}_{TjðSÞg ¼}₂_ðp jpiÞ 1þ Xr1 l¼1 logp½l !a ð1þ

c

Þ þpi 1þ Xr1 l¼1 logp½lþlogpj !a pj 1þ Xr1 l¼1 logp½lþlogpi !a P0:

Thus, {Tj(S0) +Ti(S0)}{Ti(S) +Tj(S)}P0. This completes the proof ofTheorem 4. h

Theorem 5.For the1jpj½r¼pj 1þ

Pr1

l¼1logp½l

a

; spsdjPLmaxproblem, the optimal schedule is obtained by sequencing jobs in

non-decreasing order of pi, i.e., the SPT order, if the job processing times and due dates are agreeable.

Proof.It is similar to that ofTheorem 4and is omitted. h

4. Conclusions

We consider a new scheduling model in which job deterioration and past-sequence-dependent setup times exist simul-taneously. Under the proposed model, we show that the single-machine scheduling problems to minimize the makespan, total completion time, and sum of square of completion times are polynomially solvable. In addition, we show that the prob-lems to minimize the total tardiness and maximum lateness are polynomially solvable if the processing times and due dates are agreeable.

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Acknowledgements

We are grateful to the Editor and the referees for their constructive comments on an earlier version of this paper.

References

[1] J.N.D. Gupta, S.K. Gupta, Single facility scheduling with nonlinear processing times, Comput. Ind. Eng. 14 (1988) 387–393.

[2] A.S. Kunnathur, S.K. Gupta, Minimizing the makespan with late start penalties added to processing times in a single facility scheduling problem, Eur. J. Oper. Res. 47 (1) (1990) 56–64.

[3] S. Browne, U. Yechiali, Scheduling deteriorating jobs on a single processor, Oper. Res. 38 (1990) 495–498.

[4] B. Alidaee, N.K. Womer, Scheduling with time dependent processing times: review and extensions, J. Oper. Res. Soc. 50 (1999) 711–720. [5] T.C.E. Cheng, Q. Ding, B.M.T. Lin, A concise survey of scheduling with time-dependent processing times, Eur. J. Oper. Res. 152 (2004) 1–13. [6] S. Gawiejnowicz, W. Kurc, L. Pankowska, Pareto and scalar bicriterion optimization in scheduling deteriorating jobs, Comput. Oper. Res. 33 (2006) 746–

767.

[7] M. Ji, Y. He, T.C.E. Cheng, Scheduling linear deteriorating jobs with an availability constraint on a single machine, Theor. Comput. Sci. 362 (2006) 115– 126.

[8] S. Gawiejnowicz, W. Kurc, L. Pankowska, Analysis of a time-dependent scheduling problem by signatures of deterioration rate sequences, Discrete Appl. Math. 154 (2006) 2150–2166.

[9] C.C. Wu, W.C. Lee, Two-machine flowshop scheduling to minimize mean flow time under linear deterioration, Int. J. Prod. Econ. 103 (2006) 572–584. [10] Y.R. Shiau, W.C. Lee, C.C. Wu, C.M. Chang, Two-machine flowshop scheduling to minimize mean flow time under simple linear deterioration, Int. J. Adv.

Manuf. Technol. 34 (2007) 774–782.

[11] L. Kang, C.T. Ng, A note on a fully polynomial-time approximation scheme for parallel-machine scheduling with deteriorating jobs, Int. J. Prod. Econ. 109 (2007) 180–184.

[12] S. Gawiejnowicz, Scheduling deteriorating jobs subject to job or machine availability constraints, Eur. J. Oper. Res. 180 (2007) 472–478.

[13] C.C. Wu, W.C. Lee, Y.R. Shiau, Minimizing the total weighted completion time on a single machine under linear deterioration, Int. J. Adv. Manuf. Technol. 33 (2007) 1237–1243.

[14] S. Raut, S. Swami, J.N.D. Gupta, Scheduling a capacitated single machine with time deteriorating job values, Int. J. Prod. Econ. 114 (2008) 769–780. [15] W.C. Lee, C.C. Wu, Y.H. Chung, Scheduling deteriorating jobs on a single machine with release times, Comput. Ind. Eng. 54 (2008) 441–452. [16] W.C. Lee, C.C. Wu, Y.H. Chung, H.C. Liu, Minimizing the total completion time in permutation ﬂow shop with machine-dependent job deterioration

rates, Comput. Oper. Res. 36 (2009) 2111–2121.

[17] D. Biskup, J. Herrmann, Single-machine scheduling against due dates with past-sequence-dependent setup times, Eur. J. Oper. Res. 191 (2) (2008) 587– 592.

[18] J.-B. Wang, Q. Guo, A due-date assignment problem with learning effect and deteriorating jobs, Appl. Math. Model. 34 (2010) 309–313.

[19] C.T. Ng, J.-B. Wang, T.C.E. Cheng, L.L. Liu, A branch-and-bound algorithm for solving a two-machine ﬂow shop problem with deteriorating jobs, Comput. Oper. Res. 37 (1) (2010) 83–90.

[20] T.C.E. Cheng, J.N.D. Gupta, G. Wang, A review of ﬂowshop scheduling research with setup times, Prod. Oper. Manage. 9 (3) (2000) 262–282. [21] A. Allahverdi, J.N.D. Gupta, T. Aldowaisan, A review of scheduling research involving setup considerations, Omega 27 (1999) 219–239.

[22] A. Allahverdi, C.T. Ng, T.C.E. Cheng, M.Y. Kovalyov, A survey of scheduling problems with setup times or costs, Eur. J. Oper. Res. 187 (3) (2008) 985– 1032.

[23] C. Koulamas, G.J. Kyparisis, Single-machine scheduling with past-sequence-dependent setup times, Eur. J. Oper. Res. 187 (2008) 1045–1049. [24] C.C. Wu, Y.R. Shiau, W.C. Lee, Single-machine group scheduling problems with deterioration consideration, Comput. Oper. Res. 35 (2008) 1652–1659. [25] C.C. Wu, W.C. Lee, Single-machine group-scheduling problems with deteriorating setup times and job-processing times, Int. J. Prod. Econ. 115 (2008)

128–133.

[26] W.C. Lee, C.C. Wu, Multi-machine scheduling with deteriorating jobs and scheduled maintenance, Appl. Math. Model. 32 (2008) 362–373. [27] A. Allahverdi, H.M. Soroush, The signiﬁcance of reducing setup times/setup costs, Eur. J. Oper. Res. 187 (2008) 978–984.

[28] T. Eren, A bicriteria parallel machine scheduling with a learning effect of setup and removal times, Appl. Math. Model. 33 (2009) 1141–1150. [29] J.-B. Wang, Single-machine scheduling with past-sequence-dependent setup times and time-dependent learning effect, Comput. Ind. Eng. 55 (3)

(2008) 584–591.

[30] R.L. Graham, E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling: a survey, Ann. Discrete Math. 5 (1979) 287–326.

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