Computers & Industrial Engineering 59 (2010) 663–666

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Computers & Industrial Engineering

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Single machine scheduling with past-sequence-dependent setup timesand deteriorating jobs q

Chuanli Zhao *, Hengyong TangSchool of Mathematics and Systems Science, Shenyang Normal University, Shenyang, Liaoning 110034, People’s Republic of China

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

Article history:Received 4 February 2010Received in revised form 19 July 2010Accepted 20 July 2010Available online 23 July 2010

Keywords:SchedulingSingle machineSetup timesDeteriorating jobs

0360-8352/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.cie.2010.07.015

q This manuscript was processed by Area Editor T.C* Corresponding author. Tel./fax: +86 24 86593359

E-mail address: zhaochuanli@synu.edu.cn (C. Zhao

This paper considers single machine scheduling problems with setup times and deteriorating jobs. Thesetup times are proportional to the length of the already processed jobs, that is, the setup times arepast-sequence-dependent (p-s-d). It is assumed that the job processing times are defined by functionsdependent on their starting times. The following objectives are considered: the makespan, the total com-pletion time, and the sum of earliness, tardiness, and due-window starting time and size penalties. Wepropose polynomial time algorithms to solve these problems.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Machine scheduling with deteriorating jobs has receivedincreasing attention in recent years. The first to introduce deterio-rating jobs in the context of scheduling were Gupta and Gupta(1988) and Browne and Yechiali (1990). Mosheiov (1991) studiedthe single machine scheduling problem of minimizing the totalcompletion time under linear deterioration, where all jobs hadthe same processing time. It shows that the optimal schedule isV-shaped. Mosheiov (1994) considered the single machine sched-uling problem with simple linear deterioration. His work showedthat the problems of minimizing such objectives as makespan, to-tal completion time, total weighted completion time, total latenessnumber of tardy jobs, maximum lateness and maximum tardinessare all polynomially solvable. Cheng, Kang, and Ng (2004b) studiedthe single machine scheduling problem with the same deteriora-tion rate, the objective of which was to minimize the sum ofearliness, tardiness and due-date penalties. They provided someproperties and an algorithm to solve the problem. Apart from thesearticles, there are other contributions to scheduling problems withdeteriorating jobs, see for example Alidaee and Womer (1999),Bachman, Janiak, and Kovalyov (2002), Cheng, Ding, and Lin(2004a, 2005), Zhao and Tang (2005) and Yang et al. (2010).

Moreover, it is reasonable and necessary to consider schedulingproblems with setup times (Allahverdi, Ng, Cheng, & Kovalyov,2008). In classical scheduling, there are two types of setup time:

ll rights reserved.

. Edwin Cheng..).

sequence-independent and sequence-dependent. In the first case,the setup times are usually added to the job processing times whilein the second case the setup times depends on both the job cur-rently being scheduled and the last scheduled job. Koulamas andKyparisis (2008) first introduced a scheduling problem with past-sequence-dependent (p-s-d) setup times, i.e., the setup time isdependent on all already scheduled jobs. Koulamas and Kyparisisstudied some single machine scheduling problems, the objectivesof which are minimizing the makespan, the total completion time,and the total absolute differences in completion times. Their workshowed that all these problems can be solvable in polynomial time.Some extensions with nonlinear p-s-d setup times are also consid-ered. Biskup and Herrmann (2008) considered single machinescheduling with past-sequence-dependent setup times and due-date. Their results showed that some problems remained to bepolynomially solvable even when the past-sequence-dependentsetup times were included in the analysis. Kuo and Yang (2007)investigated some single machine scheduling problems withpast-sequence-dependent setup times and learning effect. Theyproved that the makespan minimization problem, the total com-pletion time minimization problem, the total absolute differencesin completion times and the sum of earliness, tardiness, and com-mon due-date penalties minimization problem could be solved inpolynomial time, respectively. Wang (2008) extended the resultof Kuo and Yang (2007) to the single machine scheduling problemwith a time-dependent learning effect and past-sequence-depen-dent setup times. The objectives are minimizing the makespan,the total completion time and the sum of the quadratic job comple-tion times. Recently, Wang, Jiang, and Wang (2009) studied singlemachine scheduling with past-sequence-dependent setup times

664 C. Zhao, H. Tang / Computers & Industrial Engineering 59 (2010) 663–666

and effects of deterioration and learning. His study showed thatthe makespan minimization problem, the total completion timeminimization problem, and the sum of the dth power of job com-pletion times minimization problem can be solved by the SDR rule,respectively.

In this paper, we consider single machine scheduling problemswith past-sequence-dependent setup times and deteriorating jobs.

The model can be described as follows.There are n independent and non-preemptive jobs to be pro-

cessed on a single machine. All the jobs are available simulta-neously at time zero. Each job Jj has a normal processing timeaj(j = 1, . . . ,n). We assume that the actual processing time of job Jj

if it is starts at time t is given by:

pj ¼ aj þ bt ð1Þ

where b P 0 is a deteriorating index. Also, we assumed that the p-s-d setup time of J[j] when scheduled in position j is given as Koula-mas and Kyparisis (2008) did, as follows:

s½j� ¼ dXj�1

i¼1

p½i�; j ¼ 2; . . . ;n; s½1� ¼ 0 ð2Þ

where d P 0 is a normalizing constant.We denote the problem as

1jaj þ bt; SpsdjZ

2. Regular objectives

In this section, we consider the problems 1jaj + bt, SpsdjCmax and1jaj þ bt; Spsdj

PCj.

First, two useful lemmas are given.

Lemma 1. [Hardy, Littlewood, and Polya, 1967]We assume thereare two sequences of numbers xi and yi(i = 1, . . . ,n). The sum

Pni¼1xiyi

of products of the corresponding elements is the least if the sequencesare monotonic in the opposite sense.

Let p = [J[1], J[2], . . . , J[n]] is a schedule, p[j] and a[j], respectively,denote the actual processing time and the normal processing time ofjob J[j]. Then the actual processing times of jobs can be expressed asfollows:

p½1� ¼ a½1�

p½2� ¼ a½2� þ ba½1�

p½3� ¼ a½3� þ b½ð1þ bÞa½1� þ a½2��

. . . ; . . . ;

p½j� ¼ a½j� þ bXj�1

i¼1

ð1þ bÞj�i�1a½i�

" #

. . . ; . . . ;

p½n� ¼ a½n� þ bXn�1

i¼1

ð1þ bÞn�i�1a½i�

" #

If the job processing times are time-independent (i.e., b = 0), the1jaj + btjZ problem reduces to the 1kZ problem.

Lemma 2. For a given schedule p = [J[1], J[2], . . . , J[n]], if the value ofthe objective for the 1kZ problem can be formulated asZ ¼

Pnj¼1w½j�a½j�, then the value of the objective for the 1jaj + btjZ

problem can be formulated as Z ¼Pn

j¼1w½j�p½j� ¼Pn

j¼1 ~w½j�a½j�, where

~w½j� ¼ w½j� þ bPn

i¼jþ1ð1þ bÞi�j�1w½i� (we definePn

j¼nþ1xj ¼ 0).

Proof. It is obviously that the actual processing time of job J[j] is a[j]

in the 1kZ(b = 0) problem and it is p[j] in the 1jaj + btjZ problem.Since the value of the objective for the 1kZ problem can be formu-lated as Z ¼

Pnj¼1w½j�a½j�, then the value of the objective for the

1jaj + btjZ problem can be formulated as Z ¼Pn

j¼1w½j�p½j�. Therefore,

Z ¼Xn

j¼1

w½j�p½j� ¼Xn

j¼1

w½j� a½j� þ bXj�1

i¼1

ð1þ bÞj�i�1a½i�

" #( )

¼Xn

j¼1

w½j� þ bXn

i¼jþ1

ð1þ bÞi�j�1w½i�

" #a½j� ¼

Xn

j¼1

~w½j�a½j�

This completes the proof of Lemma. h

We first consider the 1jaj + bt, SpsdjCmax problem. Let p =[J[1], J[2], . . . , J[n]], then

Cmax ¼Xn

r¼1

ðs½r� þ p½r�Þ ¼Xn

r¼1

½ðn� rÞdþ 1�p½r� ¼Xn

r¼1

w½r�p½r�

¼Xn

r¼1

w½r� þ bXn

i¼rþ1

ð1þ bÞi�r�1w½i�

" #a½r� ¼

Xn

r¼1

~w½j�a½r� ð3Þ

where w[r] = (n � r)d + 1 and ~w½j� ¼ w½r� þ bPn

i¼rþ1ð1þ bÞi�r�1w½i�.From Lemma 1, Eq. (3) is minimized by sorting the elements of

the ~w½j� and a[j] vectors in opposite orders. Consequently, an opti-mal sequence for the 1jaj + bt, SpsdjCmax problem can be obtainedin O(nlogn) time.

We now turn our attention to the 1jaj þ bt; SpsdjP

Cj problem.Clearly,

Xn

j¼1

Cj ¼Xn

r¼1

ðn� r þ 1Þðs½r� þ p½r�Þ

¼Xn

r¼1

ðn� r þ 1Þ 1þ dn� r

2

� �p½r� ¼

Xn

r¼1

w½r�p½r�

¼Xn

r¼1

w½r� þ bXn

i¼rþ1

ð1þ bÞi�r�1w½r�

" #a½r� ¼

Xn

r¼1

~w½j�a½r� ð4Þ

where w½r� ¼ðn�rþ1Þð1þdn�r2 Þ and ~w½j�¼w½r�þb

Pni¼rþ1ð1þbÞi�r�1w½r�.

Similar to the 1jaj + bt, SpsdjCmax problem, from Lemma 1, Eq. (4)is minimized by sorting the elements of the ~w½j� and a[r] vectors inopposite orders. Consequently, an optimal sequence for the1jaj þ bt; Spsdj

PCj problem can be obtained in O(nlogn) time.

3. Due-window assignment problem

Liman, Panwalker, and Thongmee (1998) considered a single-machine earliness-tardiness scheduling problem with a commondue-window: jobs need to be processed around a common due-window; jobs completed within the due-window are consideredas on-time jobs, whereas early and tardy jobs are penalized. Thereare other contributions to scheduling problems with common due-date or common due-window, see for example Cheng (1988, 1989),Yeung, Oguz, and Cheng (2004, 2009) Yeung, Choi, and Cheng (inpress). In this section we extend the model in Liman et al. (1998)to include past-sequence-dependent setup times and deterioratingjobs. Assume that all the jobs share a common due-window. Let d1

and d2(d2 P d1) denote the starting time and the finishing time ofthe due-window, respectively. Let D = d2 � d1 denote its size. LetEj = max{0,d1 � Cj} and Tj = max{0,Cj � d2} be the earliness and tar-diness of job Jj(j = 1, . . . ,n), respectively. Further, let a and b repre-sent the unit penalties for earliness and tardiness respectively; crepresents the unit penalty of (delaying) the due-window startingtime, and h represents the unit penalty of (increasing) the due-win-dow size. The objective is to determine (i) the job schedule, (ii) the

C. Zhao, H. Tang / Computers & Industrial Engineering 59 (2010) 663–666 665

starting time and length of the due-window, such that the follow-ing objective function is minimized:

Zðd1; d2;pÞ ¼Xn

j¼1

ðaEj þ bTj þ cd1 þ hDÞ:

We first give several properties of an optimal solution.

Lemma 3. For the 1jaj þ bt; SpsdjPðaEj þ bTj þ cd1 þ hDÞ problem,

(i) There exists an optimal schedule in which the schedule starts attime zero and no idle time between consecutive jobs.

(ii) If c > h, an optimal schedule exists in which the due-windowstarts at time zero.

(iii) If b < min{c,h}, an optimal schedule exists in which the due-window is reduced to a due-date that starts (and is completed)at time zero.

Proof. Similar to the proof in Mosheiov and Sarig (2009). h

Lemma 4. For the 1jaj þ bt; SpsdjPðaEj þ bTj þ cd1 þ hDÞ problem

there exists an optimal schedule in which

(i) Both the due-window starting time d1 and the due-windowcompletion time d2 coincide with job completion times, and

(ii) d1 = C[k] and d2 = C[k+l], where k ¼ dnðh�cÞa e and kþ l ¼ dnðb�hÞ

b e.(iii) The optimal value of the objective function Zðd;pÞ ¼Pn

r¼1w½r�ðs½r� þ p½r�Þ, where the positional weight wr = min{a(r �1) + nc,hn,b(n � r + 1)}, r = 1, . . . ,n.

Proof

(i) We first show that there exists an optimal schedule in whichd1 coincides with a job completion time. Suppose p = [J[1],J[2], . . . , J[n]] is an optimal schedule such that C[k] 6 d1 6 C[k+1],and let Z(d1,d2,p) be the corresponding objective value. Usingthe standard technique of small perturbation, we measure thechange in the total cost when moving d1. Define D1 = d1 � C[k]

and D2 = C[k+1] � d1. Let Z(d1 � D1,d2,p) and Z(d1 + D2,d2,p)be the objective value for d1 = C[k] and d1 = C[k+1], respectively.Then Z(d1 � D1,d2,p) = Z(d1,d2,p) + D1(nh � nc � ka) andZ(d1 + D2,d2,p) = Z(d1,d2,p) � D2(nh � nc � ka).

If (nh � nc � ka) < 0, then Z(d1 � D1,d2, p) < Z(d1,d2,p). Other-wise, Z(d1 + D2,d2,p) 6 Z(d1,d2,p). Therefore, there exists an opti-mal schedule in which d1 coincides with a job completion time.

Now we show that there exists an optimal schedule in which d2

coincides with a job completion time. Suppose p = [J[1], J[2], . . . , J[n]] isan optimal schedule such that Ck] 6 d2 6 C[k+1]. Define D1 = d2 � C[k]

and D2 = C[k+1] � d2. Then Z(d1,d2 � D1,p) = Z(d1,d2,p) + D1[nh �nc � (n � k)b] and Z(d1, d2 + D2,p) = Z(d1,d2,p) � D2[nh � nc �(n � k)b].

If [nh � nc � (n � k)b] < 0, then Z(d1,d2 � D1,p) < Z(d1,d2,p).Otherwise, Z(d1,d2 + D2,p) 6 Z(d1,d2,p). Therefore, there exists anoptimal schedule in which d2 coincides with a job completion time.

(ii) We first show that d1 = C[k], where k ¼ dnðh�cÞa e. Suppose

p = [J[1], J[2], . . . , J[n]] is an optimal schedule such thatd1 = C[k]. The effect of moving d1D units of time to the left(i.e., smaller) is [�a(k � 1) � cn + hn]D. The effect of movingd1D units of time to the right (i.e., larger) is [ak + cn � hn]D.

Since d1 = C[k] is optimal due-date, then [�a(k � 1) �cn + hn] P 0 and [ak + cn � hn] P 0, it follows that k ¼ dnðh�cÞ

a e.

Similarly, we can show that d2 = C[k+l], where kþ l ¼ dnðb�hÞb e.

(iii) Suppose p = [J[1], J[2], . . . , J[n]] is an optimal schedule such thatd1 = C[k] and d2 = C[k+l], we have

Zðd1;d2;pÞ ¼ aXk�1

j¼1

E½j� þ bXn

j¼kþ1

Tj þ ncd1 þ nhD

¼ aXk�1

j¼1

Xk

i¼jþ1

ðs½i� þ p½i�Þ þ bXn

j¼kþlþ1

Xj

i¼kþ1

ðs½i� þ p½i�Þ

þ cnXk

j¼1

ðs½j� þ p½j�Þ þ hnXkþl

j¼kþ1

ðs½j� þ p½j�Þ

¼Xk

r¼1

½aðr � 1Þ þ nc�ðs½r� þ p½r�Þ þ hnXkþl

j¼kþ1

ðs½j� þ p½j�Þ

þXn

r¼kþlþ1

bðn� r þ 1Þðs½r� þ p½r�Þ

¼Xn

r¼1

w½r�ðs½r� þ p½r�Þ: ð5Þ

where wr = min{a(r � 1) + nc,hn,b(n � r + 1)}. h

Remark 1

(i) k is the largest r value such that a(r � 1) + nc 6 min{-hn,b(n � r + 1)}. If no r value satisfies this inequality, thenk = 0.

(ii) k + l is the largest r value such that hn 6min{a(r � 1) +nc,b(n � r + 1)}. If no r value satisfies this inequality, thenl = 0.

According to Lemma 4, for the 1jaj þ bt; SpsdjPðaEj þ bTjþ

cd1 þ hDÞ problem, the objective can be rewritten as:

Zðd;pÞ ¼Xn

r¼1

w½r�ðs½r� þ p½r�Þ ¼Xn

r¼1

ðw½r� þ dXn

l¼rþ1

w½l�Þp½r�

¼Xn

r¼1

~w½r�p½r� ¼Xn

r¼1

½ ~w½r� þ bXn

i¼rþ1

ð1þ bÞi�r�1 ~w½r��a½r�

¼Xn

r¼1

W ½r�a½r� ð6Þ

where w[r] = min{a(r � 1) + nc,hn,b(n � r + 1)} ~w½r� ¼ w½r� þ dPn

l¼rþ1

w½l�, and W ½r� ¼ ~w½r� þ bPn

i¼rþ1ð1þ bÞi�r�1 ~w½i�.Based on the above analysis, we propose the following O(nlogn)

algorithm to solve the 1jaj þ bt; SpsdjPðaEj þ bTj þ cd1 þ hDÞ

problem.

Algorithm 1

Step 1: Assign the optimal due-dates d1 = C[k] and d2 = C[k+l], wherek ¼ dnðh�cÞ

a e and kþ l ¼ dnðb�hÞb e.

Step 2: Calculate each value of wr, ~wr and Wr, r = 1, . . . ,n.Step 3: Assign the job with the longest normal processing time (aj)

to the position with the smallest position weight Wr, thejob with the second longer normal processing time tothe position with the second smaller position weight ofWr, etc.

4. Conclusions

In this paper we consider single machine scheduling problemswith past-sequence-dependent (p-s-d) setup time and deteriorat-

666 C. Zhao, H. Tang / Computers & Industrial Engineering 59 (2010) 663–666

ing jobs. It is assumed that the job processing times are defined byan increasing functions dependent on their starting times, i.e.,aj + bt. The objective including the makespan, the total completiontime, and the sum of earliness, tardiness, and due-window startingtime and size penalties. We propose polynomial time algorithms tosolve these problems. The algorithm can also be easily applied tothe ‘mirror’ scheduling problem in which the actual processingtime of job is given by aj � bt (the parameter b must be given prop-erly to guarantee pj > 0). Scheduling with past-sequence-depen-dent setup time and deteriorating jobs in other machine settingsare clearly interesting and significant topics for future research.

Acknowledgments

The authors wish to thank two referees for their constructivecomment and suggestions that improved an early version of thispaper. This work was supported by National Natural Science Foun-dation of China (NNSFC) 10471096.

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