Computers & Industrial Engineering 58 (2010) 326–331

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Computers & Industrial Engineering

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

Scheduling problems with deteriorating jobs and learning effects includingproportional 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 Kongb Department of Statistics, Feng Chia University, Taichung, Taiwan

a r t i c l e i n f o

Article history:Received 28 January 2009Received in revised form 25 June 2009Accepted 18 November 2009Available online 23 November 2009

Keywords:Deteriorating jobsLearningProportional setup times

0360-8352/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.cie.2009.11.008

* Corresponding author. Tel.: +852 2766 5216; fax:E-mail address: LGTcheng@polyu.edu.hk (T.C.E. Ch

a b s t r a c t

Recently, interest in scheduling with deteriorating jobs and learning effects has kept growing. However,research in this area has seldom considered setup times. We introduce a new scheduling model in whichjob deterioration and learning, and setup times are considered simultaneously. In the proposed model,the actual processing time of a job is defined as a function of the setup and processing times of the jobsalready processed and the job’s own scheduled position in a sequence. In addition, the setup times areassumed to be proportional to the actual processing times of the already scheduled jobs. We derive poly-nomial-time optimal solutions for some single-machine problems with or without the presence of certainconditions.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

In conventional scheduling models, job processing times areoften assumed known and fixed throughout the duration of jobprocessing. However, it can be found in many situations, wherethe processing times of jobs might be prolonged due to deteriora-tion or shortened due to learning over time. In the deteriorationcase, e.g., the temperature of a molten iron ingot, while waitingto enter a rolling machine, drops below a certain level and theingot needs to be reheated before rolling (Kunnathur & Gupta,1990). In such environments, a job that is processed later con-sumes more time than the same job if processed earlier. Schedulingin this setting is known as ‘‘scheduling with deteriorating jobs”,which was first independently introduced by Gupta and Gupta(1988), and Browne and Yechiali (1990). Since then, models ofscheduling with deteriorating jobs have been extensively studiedfrom a variety of perspectives. For instance, Cheng and Ding(1998), Bachman, Cheng, Janiak, and Ng (2002), Ng, Cheng, andBachman (2002), Cheng, Ding, Kovalyov, Bachman, and Janiak(2003), and Wu and Lee (2008) considered models, where the jobprocessing times are linear functions of their starting times.Alidaee and Womer (1999) surveyed the rapidly growing litera-ture, while Cheng, Ding, and Lin (2004) gave a detailed review ofscheduling problems with deteriorating jobs.

On the other hand, in the learning situation, Biskup (1999)pointed out that repeated processing of similar tasks improvesworkers’ skills, e.g., workers are able to perform setups, deal with

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+852 2364 5245.eng).

machine operations or software, or handle raw materials and com-ponents at a faster pace. Scheduling in this setting is known as‘‘scheduling with learning effects”. The pioneers that introducedthe concept of learning into the field of scheduling were Biskup(1999) and Cheng and Wang (2000). Since then, many researchershave devoted much effort to studying this nascent and vivid area ofscheduling. Recently, Biskup (2008) presented a comprehensive re-view of research on scheduling with learning effects. Eren (2009)proposed a non-linear mathematical programming model for thesingle-machine scheduling problem with unequal release datesand learning effects. Eren and Güner (2008) considered a two-ma-chine flowshop with position-based learning, where the objectiveis to minimize the weighted sum of the total completion timeand makespan. Janiak and Rudek (2009) introduced a new modelof learning into the scheduling field that relaxes one of the rigidconstraints by assuming that each job provides a different experi-ence to the processor. They formulated the shape of the learningcurve as a non-increasing k-stepwise function. Moreover, Janiakand Rudek (2008) proposed a new experience-based learning mod-el, where the job processing times are described by S-shaped func-tions dependent on the experience of the processor. Janiak et al.(2009) investigated the single-processor problem to minimizethe makespan with an S-shaped learning model. Chang et al.(2009) considered the problem of scheduling a set of n indepen-dent jobs on a single machine with a learning/aging effect.

Interest in scheduling with deteriorating jobs and learning ef-fects has been growing since Lee (2004) assumed that the actualprocessing time of a job j scheduled in position r of a sequence ispj[r] = ajtr

a or pj[r] = (p0 + ajt)ra, where aj is the rate of job deteriora-tion, t P 0 is the starting time of processing the job, a 6 0 is the

T.C.E. Cheng et al. / Computers & Industrial Engineering 58 (2010) 326–331 327

learning index, and p0 is the common basic processing time. Subse-quently, Wang (2006) assumed that the job processing times havethe form: pj[r] = (aj + bt)ra, where aj is the basic processing time andb is the common deteriorating rate. In addition, Wang (2007) stud-ied a model in which the job processing times have the form:pj[r] = pj(a(t) + bra), where pj is the basic processing time and a(t)is an increasing deterioration function with að0ÞP 0. Furthermore,J.B. Wang and Cheng (2007); X. Wang and Cheng (2007) consideredthat the processing times of jobs are defined as functions of theirstarting times and positions in a sequence, i.e., pj[r] = (pj + ajt)ra,where pj is the basic processing time and aj is the deterioration rateof job j. J.B. Wang and Cheng (2007); X. Wang and Cheng (2007)considered a model in which the actual processing time is (p0 + aj-

t)ra, where p0 is the common basic processing time, aj is the growthrate, and a is the learning index. Cheng, Wu, and Lee (2008) studieda new scheduling model with deteriorating jobs and learningeffects in which the actual processing time of a job j scheduled inthe rth position of a sequence is modelled as pj½r� ¼

pjp0þPr�1

l¼1p½l�

p0þPn

l¼1pl

� �a1

ra2 , where p[l] denotes the normal processing time

of the job scheduled in the lth position of the sequence, p0 > 0 isa given parameter, and a1 and a2 denote the deteriorating andlearning indices with a1 < 0 and a2 < 0.

The above works on scheduling with deteriorating jobs andlearning effects neglected the setup cost or setup times. However,scheduling with setup times or setup cost plays a crucial role in to-day’s manufacturing and service environments, where reliableproducts/services are to be delivered on time. For example, Koula-mas and Kyparisis (2008) pointed out that in high-tech manufac-turing jobs are commonly processed in batches, e.g., a batch ofjobs may consist of a group of electronic components mounted to-gether on an integrated circuit (IC) board, whereby each batch in-curs a setup time. In addition to this un-readiness of components,Biskup and Herrmann (2008) provided another example of wear-out of equipment (e.g., a drill), in which the sum of the processingtimes of the prior jobs adds to the processing time of the actual job.Allahverdi, Ng, Cheng, and Kovalyov (2008) claimed that setupactivities due to changeovers represent costly disruptions to pro-duction/service processes. Therefore, setup reduction is an impor-tant feature of the continuous improvement programme of anymanufacturing/service organization. It is especially critical if anorganization seeks to make timely responses to market changesthrough shortened lead times and production in smaller lot sizes.Every scheduler should understand the principles of setup reduc-tion and be able to recognize its potential benefits. Motivated bythis observation, we introduce a new scheduling model in whichthe actual processing time of a job depends not only on the totalnormal processing times of the jobs already processed, but alsoon its scheduled position, and the setup times are further assumedto be proportional to the actual processing times of the alreadyscheduled jobs.

The remainder of this paper is organized as follows. We intro-duce the notation and problem formulation in the next section.In Section 3 we derive the optimal solutions for some single-ma-chine problems. We conclude the paper in the final section.

2. Notation and problem formulation

Before presenting the main results, we introduce some notationthat will be used throughout the paper.n the number of jobspj the normal processing time of job jdj the due date of job jpj½r� the actual processing time of job j scheduled in the rth

position

p½r� the normal processing time of a job scheduled in the rthposition

sj½r� the past-sequence-dependent setup time of job j scheduledin the rth position

Cj the completion time of job jLj the lateness of job j, i.e., Lj = Cj � dj

Tj the tardiness of job j, i.e., Tj = max {Lj, 0}Cmax the makespan, i.e., Cmax ¼maxfCj j ¼ 1;2; . . . ;nj gP

Cj the total completion time of all the jobsPC2

j the sum of the quadratic job completion times of all thejobs

Lmax the maximum lateness of all the jobs, i.e.,Lmax ¼maxfLjjj ¼ 1;2; . . . ;ngP

Tj the total tardiness of all the jobsc the normalizing constant number with 0 < c < 1c0; c1;m1;m2 four numbers, where c0 ¼ 1

1þPr�1

l¼1log p½l�

, c1 ¼ rþ1r

� �a2 ,

m1 ¼ 1c0

1� cþ1c1

� �and m2 ¼ logðcþ1Þ

log 2

where 0 < c0 < 1, c1 P 1, m1 < 0, and m2 > 0; a1 and a2 are the learningand deterioration indices, respectively, where m1 < a1 < 0, 0 < a2 < m2.

The formulation of the proposed problem is as follows. Thereare n jobs ready to be processed on a single machine. Preemptionis not allowed and the machine is only able to process one job ata time. Each job j has a normal processing time pj and a due datedj. We model the learning effect based on the observation of Wang(2008) that the processing times of the jobs processed before a jobcontribute to the actual processing time of a job. However, the ac-tual processing time of a given job declines to zero dramatically asthe job size increases or the normal processing time is large in bothmodels. To overcome this unrealistic situation, we assume that theactual processing time of a job is a function of the sum of the log-arithm of the normal processing times of the jobs processed beforeit. Thus, if a job j is scheduled in the rth position of a sequence, thenits actual processing time is

pj½r� ¼ pj 1þXr�1

l¼1

log p½l�

!a1

ra2 ;

where p[l] denotes the normal processing time of the job scheduledin the lth position of the sequence, and a1 and a2 denote the learningand deterioration indices.

We also take setup times into consideration in the schedulingmodel by adopting the notation of Koulamas and Kyparisis(2008) and Wang (2008) that setup times are past-sequence-depen-dent (p-s-d). Koulamas and Kyparisis (2008) motivated theassumption of p-s-d setup times by the manufacturing of inte-grated circuit (IC) boards. Specifically, an IC board consists of anumber of electronic components mounted on it and the process-ing of any electronic component will have an adverse effect on the‘‘readiness” of other components on the board due to the passageof electricity through the board. Consequently, each componentprior to processing requires a setup operation to restore it to the‘‘full-readiness” status, and the setup time depends on the compo-nent’s degree of ‘‘un-readiness”, which is proportional to the actualprocessing times of the already processed components. Thus, thep-s-d setup time of job j if it is scheduled in the rth position of asequence is as follows:

sj½1� ¼ 0 and sj½r� ¼ cXr�1

l¼1

p½l�;

where c is a normalizing constant number with 0 < c < 1, and p[l] de-notes the actual processing time of a job if it is scheduled in the lthposition.

For notational convenience, we denote all the problems underconsideration using the three-field notation scheme ajbjd for

328 T.C.E. Cheng et al. / Computers & Industrial Engineering 58 (2010) 326–331

scheduling problems introduced by Graham, Lawler, Lenstra, andRinnooy Kan (1979).

3. Some single-machine results

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

Lemma 1. cþ1þa1c0c1ð1þ c0xÞa1�1�c1ð1þc0xÞa1 P 0 for 0 < c <1,0 < c0 < 1, c1 > 1, x P 0 and m1 < a1 < 0.

Proof. Let hðxÞ ¼ cþ 1þ a1c0c1ð1þ c0xÞa1�1 � c1ð1þ c0xÞa1 . Takingthe first derivative of h(x) with respect to x, we have

h0ðxÞ ¼ a1ða1 � 1Þc20c1ð1þ c0xÞa1�2 � a1c0c1ð1þ c0xÞa1�1

:

Since m1 < a1 < 0, 0 < c0 < 1, c1 > 1 and x P 0, h(x) is an increas-ing function for x P 0. Since hðxÞP hð0Þ ¼ cþ 1þ a1c0c1 � c1 for0 < c < 1, m1 < a1 < 0, 0 < c0 < 1 and c1 > 1, we have

hðxÞP 0

for 0 < c < 1, m1 < a1 < 0, 0 < c0 < 1 and c1 > 1, and x P 0. This com-pletes the proof. h

Lemma 2. ðcþ1Þðh�1Þþc1ð1þc0 loghþc0xÞa1 �c1hð1þc0xÞa1 P 0for 0 < c < 1, h P 1, c1 > 1, 0 < c0 < 1, x > 0 and m1 < a1 < 0.

Proof. Let f ðhÞ ¼ ðcþ 1Þðh� 1Þ þ c1ð1þ c0 log hþ c0xÞa1 � c1hð1þc0xÞa1 . Taking the first and second derivatives of f(h) with respectto h, we have

f 0ðhÞ ¼ ðcþ 1Þ þ a1c0c1ð1þ c0 log hþ c0xÞa1�1

h� c1ð1þ c0xÞa1

and

f 00ðhÞ ¼ a1ða1 � 1Þc20c1ð1þ c0 log hþ c0xÞa1�2

h2

� a1c0c1ð1þ c0 log hþ c0xÞa1�1

h2 :

Since h P 1, c1 > 1, 0 < c0 < 1, x > 0 and m1 < a1 < 0, f 00ðhÞ > 0.Therefore, f0(h) is a non-decreasing function for h P 1. FromLemma 1, we have

f 0ð1Þ ¼ cþ 1þ a1c0c1ð1þ c0xÞa1�1 � c1ð1þ c0xÞa1 P 0:

Using the fact that f0(h) is a non-decreasing function for h P 1,we see that f 0ðhÞP f 0ð1ÞP 0. Therefore, we obtain that f(h) is anon-decreasing function for h P 1. Since f(1) = 0, we have

f ðhÞP 0

for 0 < c < 1, h P 1, c1 > 1, 0 < c0 < 1, x > 0 and m1 < a1 < 0. This com-pletes the proof. h

Theorem 1. For the 1=pj½r� ¼ pj 1þPr�1

l¼1 log p½l�� �a1

ra2 ; spsd=Cmax

problem, the optimal schedule is obtained by sequencing jobs in theshortest processing time (SPT) order.

Proof. Suppose pi 6 pj. Let S and S0 be two job schedules, wherethe difference between S and S0 is a pairwise interchange of twoadjacent jobs i and j, i.e., S ¼ ðp i j p0Þ and S0 ¼ ðp j i p0Þ, where pand p0 denote partial sequences. Furthermore, we assume thatthere are r � 1 jobs in p. Thus, jobs i and j are the rth and(r + 1)th jobs in S, respectively, whereas jobs j and i are scheduledin the rth and (r + 1)th positions in S0, respectively. In addition, let Adenote the completion time of the last job in p. To show that Sdominates S0, it suffices to show that the (r + 1)th jobs in S and S0

satisfy the condition CjðSÞ 6 CiðS0Þ. By definition, the actual pro-cessing times of job j in S and job i in S0 are given by

CjðSÞ ¼ Aþ pi 1þXr�1

l¼1

log p½l�

!a1

ra2 þ cXr�1

l¼1

p½l�

þ pj 1þXr�1

l¼1

log p½l� þ log pi

!a1

ðr þ 1Þa2

þ cXr�1

l¼1

p½l� þ pi 1þXr�1

l¼1

log p½l�

!a1

ra2

!ð1Þ

and

CiðS0Þ ¼ Aþ pj 1þXr�1

l¼1

log p½l�

!a1

ra2 þ cXr�1

l¼1

p½l�

þ pi 1þXr�1

l¼1

log p½l� þ log pj

!a1

ðr þ 1Þa2

þ cXr�1

l¼1

p½l� þ pj 1þXr�1

l¼1

log p½l�

!a1

ra2

!: ð2Þ

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

CiðS0Þ � CjðSÞ ¼ ðcþ 1Þðpj � piÞ 1þXr�1

l¼1

log p½l�

!a1

ra2

þ pi 1þXr�1

l¼1

log p½l� þ log pj

!a1

ðr þ 1Þa2

� pj 1þXr�1

l¼1

log p½l� þ log pi

!a1

ðr þ 1Þa2 : ð3Þ

On substituting h = pj/pi, c0 ¼ 1 1þPr�1

l¼1 log p½l�� �.

, c1 ¼ rþ1r

� �a2

and x = log pi into (3), and simplifying, we obtain

CiðS0Þ � CjðSÞ ¼ pi 1þXr�1

l¼1

log p½l�

!a1

ra2 ½ðcþ 1Þðh� 1Þ

þ c1ð1þ c0 log hþ c0xÞa1 � c1hð1þ c0xÞa1 �: ð4Þ

From Lemma 2, and h P 1, 0 < c0 < 1, c1 > 1, x P 0, m1 < a1 < 0,0 < c < 1 and 0 < a2 < m2, we have CiðS0Þ � CjðSÞP 0.

Thus, S dominates S0. Therefore, repeating this interchangeargument for all the jobs not sequenced in the SPT order will yieldthe desired result. h

Theorem 2. For the 1=pj½r� ¼ pj 1þPr�1

l¼1 log p½l�� �a1

ra2 ; spsdP

Cj�

problem, the optimal schedule is obtained by sequencing jobs in theSPT order.

Proof. Suppose that pi 6 pj. Let S and S0 be two job schedules,where the difference between S and S0 is a pairwise interchangeof two adjacent jobs i and j, i.e., S ¼ ðp i j p0Þ and S0 ¼ ðp j i p0Þ,where p and p0 denote partial sequences. Furthermore, we assumethat there are r � 1 jobs in p. Thus, jobs i and j are the rth and(r + 1)th jobs in S, respectively, whereas jobs j and i are scheduledin the rth and (r + 1)th positions in S0, respectively. In addition, let Adenote the completion time of the last job in p. To show that Sdominate S0, it suffices to show that the (r + 1)th jobs in S and S0 sat-isfy the conditions Cj(S) < Ci(S0) and CiðSÞ þ CjðSÞ 6 CjðS0Þ þ CiðS0Þ. Bydefinition, the actual processing times of jobs i and j in S are givenby

CiðSÞ ¼ Aþ pi 1þXr�1

l¼1

log p½l�

!a1

ra2 þ cXr�1

l¼1

p½l� ð5Þ

and

T.C.E. Cheng et al. / Computers & Industrial Engineering 58 (2010) 326–331 329

CjðSÞ ¼ Aþ pi 1þXr�1

l¼1

log p½l�

!a1

ra2 þ cXr�1

l¼1

p½l�

þ pj 1þXr�1

l¼1

log p½l� þ log pi

!a1

ðr þ 1Þa2

þ cXr�1

l¼1

p½l� þ pi 1þXr�1

l¼1

log p½l�

!a1

ra2

!: ð6Þ

Similarly, the actual processing times of jobs i and j in S0 are

CjðS0Þ ¼ Aþ pj 1þXr�1

l¼1

log p½l�

!a1

ra2 þ cXr�1

l¼1

p½l�; ð7Þ

CiðS0Þ ¼ Aþ pj 1þXr�1

l¼1

log p½l�

!a1

ra2 þ cXr�1

l¼1

p½l�

þ pi 1þXr�1

l¼1

log p½l� þ log pj

!a1

ðr þ 1Þa2

þ cXr�1

l¼1

p½l� þ pj 1þXr�1

l¼1

log p½l�

!a1

ra2

!: ð8Þ

Since pj � pi P 0, we have Cj(S) < Ci(S0) from Theorem 1. There-fore, we only need to show that CiðSÞ þ CjðSÞ 6 CjðS0Þ þ CiðS0Þ.

From (9)–(12), we have

fCjðS0Þ þ CiðS0Þg � fCiðSÞ þ CjðSÞg

¼ ðpj � piÞ 1þXr�1

l¼1

log p½l�

!a1

ra2

þ ðcþ 1Þðpj � piÞ 1þXr�1

l¼1

log p½l�

!a1

ra2

þ pi 1þXr�1

l¼1

log p½l� þ log pj

!a1

ðr þ 1Þa2

� pj 1þXr�1

l¼1

log p½l� þ log pi

!a1

ðr þ 1Þa2 : ð9Þ

Since pj � pi P 0 and 1þPr�1

l¼1 log p½l�� �a1

ra2 > 0, the first termon the right-hand-side of (9) is non-negative. From the proof ofTheorem 1, we know that the sum of the last three terms of onthe right-hand-side of (9) is non-negative, too. Therefore, it followsthat

CiðSÞ þ CjðSÞ 6 CjðS0Þ þ CiðS0Þ:

Thus, repeating this interchange argument for all the jobs notsequenced in the SPT order will yield the desired result. h

Theorem 3. For the 1=pj½r� ¼ pj 1þPr�1

l¼1 log p½l�� �a1

ra2 ; spsdP

C2j

.problem, the optimal schedule is obtained by sequencing jobs in theSPT order.

Proof. It is similar to that of Theorem 2 and is omitted. h

In the following, we show that the earliest due date (EDD) rule pro-vides the optimal solution for the problem to minimize the total tar-diness and the maximum lateness if the job processing times and thedue dates are agreeable, i.e., di 6 dj implies pi 6 pj for all jobs i and j.

Theorem 4. For the 1=pj½r� ¼ pj 1þPr�1

l¼1 log p½l�� �a1

ra2 ; spsdP

Ti=

problem, the optimal schedule is obtained by sequencing jobs innon-decreasing order of di if the job processing times and the due datesare agreeable.

Proof. Suppose that di 6 dj, which implies pi 6 pj. The total tardi-ness of the first r � 1 jobs are the same since they are processedin the same order. Since the makespan is minimized by the SPTrule (Theorem 1), the total tardiness of partial sequence p0 in S willnot be greater than that of partial sequence p0 in S0. Thus, to provethat the total tardiness of S is less than or equal to that of S0, it suf-fices to show that TiðSÞ þ TjðSÞ 6 TjðS0Þ þ TiðS0Þ.

From (5)–(8), we derive that the tardiness of jobs i and j in S are

TiðSÞ ¼max Aþ pi 1þXr�1

l¼1

log p½l�

!a1

ra2 þ cXr�1

l¼1

p½l� � di;0

( );

and

TjðSÞ ¼ max Aþ pi 1þXr�1

l¼1

log p½l�

!a1

ra2 þ cXr�1

l¼1

p½l�

(

þpj 1þXr�1

l¼1

log p½l� þ log pi

!a1

ðr þ 1Þa2

þcXr�1

l¼1

pA½l� þ pi 1þ

Xr�1

l¼1

log p½l�

!a1

ra2

!� dj; 0

):

Similarly, the tardiness of jobs i and j in S0 are

TjðS0Þ ¼ max Aþ pj 1þXr�1

l¼1

log p½l�

!a1

ra2 þ cXr�1

l¼1

p½l� � dj; 0

( );

and

TiðS0Þ ¼ max Aþ pj 1þXr�1

l¼1

log p½l�

!a1

ra2 þ cXr�1

l¼1

p½l�

(

þpi 1þXr�1

l¼1

log p½l� þ log pj

!a1

ðr þ 1Þa2

þcXr�1

l¼1

p½l� þ pj 1þXr�1

l¼1

log p½l�

!a1

ra2

!� di;0

):

To compare the total tardiness of jobs i and j in S and in S0, weconsider two cases. In the first case, where Aþ pj

1þPr�1

l¼1 log p½l�� �a1

ra2 þ cPr�1

l¼1 p½l� 6 dj, the total tardiness of jobs i

and j in S and in S0 are

TiðSÞþTjðSÞ¼max Aþpi 1þXr�1

l¼1

logp½l�

!a1

ra2

(þcXr�1

l¼1

p½l� �di;0

)

þmax Aþpi 1þXr�1

l¼1

logp½l�

!a1

ra2

(

þcXr�1

l¼1

p½l� þpj 1þXr�1

l¼1

logp½l� þ logpi

!a1

ðrþ1Þa2

þcXr�1

l¼1

p½l� þpi 1þXr�1

l¼1

logp½l�

!a1

ra2

!�dj;0

)

and

TjðS0Þ þ TiðS0Þ ¼ max Aþ pj 1þXr�1

l¼1

log p½l�

!a1

ra2 þ cXr�1

l¼1

p½l�

(

þpi 1þXr�1

l¼1

log p½l� þ log pj

!a1

ðr þ 1Þa2

þcXr�1

l¼1

p½l� þ pj 1þXr�1

l¼1

log p½l�

!a1

ra2

!� di; 0

):

330 T.C.E. Cheng et al. / Computers & Industrial Engineering 58 (2010) 326–331

Suppose that neither Ti(S) nor Tj(S) is zero. Note that this is themost restrictive case since it includes the case that either one orboth Ti(S) and Tj(S) are zero. From Theorem 1 and the fact thatdi 6 dj, we have

fTjðS0Þ þ TiðS0Þg � fTiðSÞ þ TjðSÞg

¼ ðcþ 1Þðpj � piÞ 1þXr�1

l¼1

log p½l�

!a1

ra2

þ pi 1þXr�1

l¼1

log p½l� þ log pj

!a1

ðr þ 1Þa2

� pj 1þXr�1

l¼1

log p½l� þ log pi

!a1

ðr þ 1Þa2 þ dj

� pi 1þXr�1

l¼1

log p½l�

!a1

ra2 � cXr�1

l¼1

log p½l� � A P 0:

Thus, fTjðS0Þ þ TiðS0Þg � fTiðSÞ þ TjðSÞgP 0 in the first case. In the

second case, where Aþ pj 1þPr�1

l¼1 log p½l�� �a1

ra2 þ cPr�1

l¼1 p½l� > dj,

the total tardiness of jobs i and j in S and in S0 are

TiðSÞþTjðSÞ¼max Aþpi 1þXr�1

l¼1

logp½l�

!a1

ra2 þcXr�1

l¼1

p½l� �di;0

( )

þmax Aþpi 1þXr�1

l¼1

logp½l�

!a1

ra2 þcXr�1

l¼1

p½l�

(

þpj 1þXr�1

l¼1

logp½l� þ logpi

!a1

ðrþ1Þa2

þcXr�1

l¼1

p½l� þpi 1þXr�1

l¼1

logp½l�

!a1

ra2

!�dj;0

)

and

TjðS0Þ þ TiðS0Þ ¼ 2Aþ 2pj 1þXr�1

l¼1

log p½l�

!a1

ra2

þ pi 1þXr�1

l¼1

log p½l� þ log pj

!a1

ðr þ 1Þa2 � di � dj

� cpj 1þXr�1

l¼1

log p½l�

!a1

ra2 þ 2cXr�1

l¼1

log p½l�:

Suppose that neither Ti(S) nor Tj(S) is zero. From Theorem 1 andpi 6 pj, we have

fTjðS0Þ þ TiðS0Þg � fTiðSÞ þ TjðSÞg

¼ 2ðpj � piÞ 1þXr�1

l¼1

log p½l�

!a1

ra2 ð1þ cÞ

þ pi 1þXr�1

l¼1

log p½l� þ log pj

!a1

ðr þ 1Þa2

� pj 1þXr�1

l¼1

log p½l� þ log pi

!a1

ðr þ 1Þa2 P 0:

Thus, fTjðS0Þ þ TiðS0Þg � fTiðSÞ þ TjðSÞgP 0 in the second case.This completes the proof of Theorem 4. h

Theorem 5. For the 1=pj½r� ¼ pj 1þPr�1

l¼1 log p½l�� �a1

ra2 ; spsd Lmax=

problem, the optimal schedule is obtained by sequencing jobs innon-decreasing order of di if the job processing times and the due datesare agreeable.

Proof. It is similar to that of Theorem 4 and is omitted. h

4. Conclusions

We considered a new scheduling model in which job deteriora-tion, learning, and past-sequence-dependent setup times existsimultaneously. Under the proposed model, we showed that thesingle-machine scheduling problems to minimize the makespan,total completion time, and sum of square of completion timesare polynomially solvable. In addition, we showed that the prob-lems to minimize the total tardiness and maximum lateness arepolynomially solvable if the processing times and the due datesare agreeable. Finally, analysis of the above problems with morethan one machine is an interesting issue for future research.

Acknowledgement

We are thankful to the Editor and two anonymous referees fortheir helpful comments on an earlier version of our paper. Thispaper was supported in part by the NSC under Grant No. NSC97-2221-E-060-MY2.

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