Information Processing Letters 112 (2012) 835–838

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Information Processing Letters

www.elsevier.com/locate/ipl

Single-machine scheduling with past-sequence-dependent delivery timesand release times

Ming Liu a, Feifeng Zheng b,∗, Chengbin Chu d, Yinfeng Xu c

a School of Economics & Management, Tongji University, Shanghai, 200092, PR Chinab Glorious Sun School of Business and Management, Donghua University, Shanghai, 200051, PR Chinac School of Management, Xi’an Jiaotong University, Shannxi, 710049, PR Chinad Laboratoire Génie Industriel, Ecole Centrale Paris, Grande Voie des Vignes, 92295 Châtenay-Malabry Cedex, France

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

Article history:Received 24 October 2011Received in revised form 4 July 2012Accepted 10 July 2012Available online 23 July 2012Communicated by J. Xu

Keywords:SchedulingPast-sequence-dependent delivery timeSingle-machineRelease time

This paper studies the problem of single-machine scheduling with past-sequence-depen-dent delivery times, which was introduced in Koulamas and Kyparisis (2010) [5]. We focuson the scenario with release times such that any job is available for processing on orafter its specific release time. Both preemptive and non-preemptive models are considered,aiming at minimizing the total completion time. An optimal algorithm is presented for thepreemptive model where any job may be preempted during processing on the machineand then resumed from where it was interrupted later on. For the non-preemptive model,we show that it is NP-hard and mainly develop an approximation algorithm.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

In most industries, the manufacturing environment hasa great influence on the treatment of jobs. Recent em-pirical studies in some industries have demonstrated thatthe waiting time of a job before processing may have anadverse effect on the processing time of the job (referto Browne and Yechiali [2]; Koulamas and Kyparisis [4];Koulamas and Kyparisis [5]). In electronic manufacturingindustry, for example, an electronic component may beexposed to an electromagnetic or radioactive field whilewaiting in the pre-processing area. After its processing onone machine but before delivery to the customer, the com-ponent is required to be “treated”, e.g., in a chemical solu-tion, to remove the exposure effect from the electromag-netic/radioactive field (refer to Koulamas and Kyparisis [5]).

In literature, there are three models considering thewaiting time-induced adverse effect on the processing ofa job. The first model with deterioration effect was intro-duced by Browne and Yechiali [2]. In this model any job is

* Corresponding author.E-mail address: zhengff@mail.xjtu.edu.cn (F. Zheng).

0020-0190/$ – see front matter © 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.ipl.2012.07.002

with a so-called deteriorating processing time such that theprocessing time of a job increases in its waiting time. Inthe second model originated from Koulamas and Kypari-sis [4], each job is with a psd (past-sequence-dependent)setup time, which is used to remove the adverse effect priorto the processing of the job via a setup operation. The thirdmodel with psd delivery times was introduced by Koulamasand Kyparisis [5], in which the adverse effect of waitingdoes not impede the schedule of job processing on onemachine and shall be removed immediately after the com-pletion of processing. The time consumed to remove theadverse effect for each job is called the job’s psd deliverytime.

Different from the traditional assumption with a job-specific constant delivery time in scheduling literature [7],Koulamas and Kyparisis [5] assumed that the psd deliv-ery time of a job is proportional to the job’s waitingtime, i.e., the start time of processing. They focused on thecase with a single-machine, and proved that the problem1|qpsd|Cmax can be solved in O (n) time by simply arrangingthe longest job to the last. They also proved that the prob-lems 1|qpsd|Lmax, 1|qpsd|Tmax and 1|qpsd|

∑U j are polyno-

mially solvable since these problems can be reduced to the

836 M. Liu et al. / Information Processing Letters 112 (2012) 835–838

corresponding problems without psd delivery times by ap-propriate transformations.

In this paper we focus on single-machine schedulingunder the third model with psd delivery times. Motivatedby the phenomena in practice such that all the jobs are notready for processing at the beginning and they arrive overtime due to the limitation of supplying or storage abil-ity, we introduce release times of jobs into the model. Tothe best of our knowledge, there are no results on single-machine scheduling with psd delivery times and releasetimes in literature. The remaining of the paper is organizedas follows. In Section 2, we formally describe the problemconsidered. Section 3 presents an optimal algorithm for thepreemptive model of the problem, and Section 4 derivesan approximation algorithm for the non-preemptive model.Finally the paper is concluded in Section 5.

2. Problem description and notations

There is a single machine to process a set of n jobsJ1, J2, . . . , Jn . Each job J j (1 � j � n) is released and be-comes available at time r j . Denote by p j the processingtime of job J j , i.e., the time for processing the job on themachine.

Given a job processing schedule, denote by S j , C ′j the

start time and end time of processing J j on the machine,respectively. Notice that in the preemptive model, since J jmay be preempted and resumed for one or more times,S j represents the first time to start processing the job onthe machine. In the environment with psd (past-sequence-dependent) delivery times, the processing of J j is followedimmediately by its psd delivery treatment. Denote by q jthe psd delivery time of job J j . Notice that the psd deliverytreatment does not occupy the machine, and as mentionedbefore it has no influence on the schedule of job process-ing. We assume as in Koulamas and Kyparisis [5] that q j isproportional to the job’s start time S j . More precisely, q jis formulated as

q j = γ S j, j = 1, . . . ,n, (1)

where γ � 0 is a constant. Let C j be the completion timeof job J j , i.e., the end time of the job’s psd delivery treat-ment. Then C j = C ′

j + q j . For non-preemptive model, wehave C ′

j = S j + p j , and then

C j = C ′j + q j = S j + p j + q j

= (1 + γ )S j + p j, j = 1, . . . ,n. (2)

For preemptive model, since job J j may be preempted dur-ing processing for one or more times, let Sl

j , plj be the last

time to start processing J j and the length of time intervalfor its last continuous processing on the machine, respec-tively. That is C ′

j = Slj + pl

j . In this model,

C j = C ′j + q j = Sl

j + plj + q j

= Slj + pl

j + γ S j, j = 1, . . . ,n. (3)

For both preemptive and non-preemptive models, theobjective is to minimize the total completion time. De-note by qpsd and prmp the model with psd delivery

times and preemptions, respectively. Using the methodof three-field notation [3], we denote the preemptiveand non-preemptive models by 1|r j,prmp,qpsd|

∑C j and

1|r j,qpsd|∑

C j respectively.

3. Preemptive model 1|r j,prmp,qpsd|∑ C j

In this section we consider the preemptive model1|r j,prmp,qpsd|

∑C j . The following lemma shows that

each job shall be started for processing at its release timein an optimal schedule.

Lemma 1. In an optimal schedule of 1|r j,prmp,qpsd|∑

C j , thefirst start time of processing for each job is exactly its releasetime, that is S j = r j for 1 � j � n.

Proof. By contradiction. Suppose otherwise in an optimalschedule π , there exists at least one job J j for some 1 �j � n with S j > r j . Construct another preemptive sched-ule π ′ by inserting the processing of J j at time r j witha time length of zero. We observe that such an insertionof zero time length processing makes no change on theprocessing schedule of job J j as well as all the other jobsin π . Hence, the completion times of any job except J j inthe two schedules π and π ′ are the same. For J j , S j = r j

in π ′ due to the insertion of processing while S j > r j

in π . We conclude that the completion time of J j in π ′is strictly less than that in π by Formula (3). Hence, thetotal completion time in π ′ is strictly less than that in π ,contradicting the fact that π is an optimal schedule. Thelemma follows. �

Based on the above lemma, we propose an optimal al-gorithm named MRSPT (Modified Shortest Remaining Pro-cessing Time) which is formally described below.

Algorithm MSRPT:At any time when there releases a job J j , preemptthe currently processing job and start to process J j onthe machine with a time length of zero. The algorithmprocesses jobs by SRPT (Shortest Remaining ProcessingTime) rule which processes a job with the shortest re-maining processing time among all released jobs. Tiesare broken arbitrarily.

Below we show that algorithm MSRPT produces an op-timal schedule. By Lemma 1 and Formula (3), the com-pletion time of any job J j is equal to C j = C ′

j + q j =C ′

j + γ r j . Since γ and r j are extraneously given values,minimizing C j is equivalent to minimizing C ′

j . We con-clude that the model 1|r j,prmp,qpsd|

∑C j reduces to the

classical preemptive problem without psd delivery times,i.e., 1|r j,prmp|∑ C j , which can be solved by SRPT rule [1].Hence, we have the following theorem.

Theorem 1. Algorithm MSRPT is optimal for 1|r j,prmp,

qpsd|∑

C j .

M. Liu et al. / Information Processing Letters 112 (2012) 835–838 837

4. Non-preemptive model 1|r j,qpsd|∑ C j

This section focuses on the non-preemptive model1|r j,qpsd|

∑C j . For the corresponding problem without re-

lease times, i.e., 1|qpsd|∑

C j , Koulamas and Kyparisis [5]proved that SPT (Shortest Processing Time) rule is optimal.However, the introduction of release times of jobs makesthe problem much more complex. We claim that prob-lem 1|r j,qpsd|

∑C j is NP-hard due to the NP-hardness of

problem 1|r j |∑ C j (refer to Lenstra et al. [6]). The latterproblem is a special case of 1|r j,qpsd|

∑C j with γ = 0.

Below we propose an approximation algorithm namedCONVERT for 1|r j,qpsd|

∑C j . The main idea of the algo-

rithm is to transform an optimal preemptive schedule intoa feasible non-preemptive schedule. The total relaxation ofjobs’ completion times during the transformation will bebounded from above.

For any an input job instance, let P be an optimalpreemptive schedule that is produced by MSRPT rule. LetSl

j(P ), C ′j(P ) be the last start time and end time of pro-

cessing J j respectively, and C j(P ) be the completion timeof J j in P . Denote by S j(N) the start time of process-ing J j and by C j(N) the completion time of the job inschedule N . Below is the formal description of algorithmCONVERT which converts schedule P into a feasible non-preemptive schedule N for problem 1|r j,qpsd|

∑C j .

Algorithm CONVERT:Input: Preemptive schedule P produced by MSRPT algo-rithmOutput: Non-preemptive schedule NStep 1. Form a list L of jobs J1, J2, . . . , Jn in theincreasing order of C ′

j(P ) or equally Slj(P ). That is,

Slj(P ) < Sl

j+1(P ) for 1 � j < n in L.Step 2. Produce a non-preemptive and feasible scheduleof the n jobs in the same order as in L via the follow-ing two substeps.Step 2.1 Form an infeasible schedule L′ such that eachjob J j is started processing at time Sl

j(P ), and the ma-chine goes through processing the job during time in-terval [Sl

j(P ), Slj(P ) + p j].

Step 2.2 Postpone the start times of jobs in schedule L′ ,if necessary, in the following way. For j = 1,2, . . . ,n, ifat time Sl

j(P ) the machine has been assigned in L′ toprocess some previous job Jk for 1 � k < j, postponethe start time of J j from Sl

j(P ) onward to an earliest

available time point, that is S j(N) = Sk(N) + ∑ j−1i=k pi .

Otherwise set S j(N) = Slj(P ).

We make an explanation on the definition of S j(N) =Sk(N) + ∑ j−1

i=k pi in Step 2.2 of algorithm CONVERT. If themachine is busy in processing Jk for some 1 � k < j attime Sl

j(P ), i.e., Slj(P ) ∈ [Sk(N), Sk(N)+ pk), then we claim

that all the jobs Jk+1, . . . , J j−1 have already been releasedby time Sl

j(P ) due to the construction of job list L inStep 1 of the algorithm. Thus jobs Jk+1, . . . , J j−1, J j mustbe processed consecutively following Jk in N , implyingS j(N) = Sk(N) + ∑ j−1 pi .

i=k

Lemma 2. Given an optimal preemptive schedule P by MSRPTalgorithm, CONVERT produces in O (n log n) time a non-preemptive schedule N in which C j(N) � 2(1 + γ )C j(P ) for1 � j � n.

Proof. Consider an arbitrary job J j for 1 � j � n in thenon-preemptive schedule N . By Step 2.2 of algorithm CON-VERT, either S j(N) = Sk(N) + ∑ j−1

i=k pi for some 1 � k < j

or S j(N) = Slj(P ). For the first case, combining Sk(N) �

Slj(P ) with k � 1,

S j(N) = Sk(N) +j−1∑

i=k

pi � Slj(P ) +

j−1∑

i=1

pi

� Slj(P ) + (

C ′j(P ) − p j

)

= Slj(P ) + (

C j(P ) − γ r j − p j)< 2C j(P ) − p j,

where in the above the second inequality holds since thesingle machine is at most kept busy in processing jobsJ1, J2, . . . , J j within time interval [0, C ′

j(P )], and the third

inequality is due to Slj(P ) < C j(P ) and γ r j � 0. For the

other case where S j(N) = Slj(P ), it is trivial that S j(N) =

Slj(P ) < 2C j(P ) − p j . Together with formula (2),

C j(N) = (1 + γ )S j(N) + p j

< (1 + γ )(2C j(P ) − p j

) + p j

� 2(1 + γ )C j(P ).

The time complexity of algorithm CONVERT is as follows.Step 1 of the algorithm takes O (n log n) time to producesschedule L. Step 2.1 takes O (n) time to produces a non-feasible schedule L′ , and Step 2.2 also takes O (n) time toadjust the processing time interval of each job in sched-ule L′ . Thus CONVERT runs in O (n log n) time. The lemmafollows. �

The above lemma directly deduces the following corol-lary.

Corollary 1.∑n

j=1 C j(N) � 2(1 + γ )∑n

j=1 C j(P ).

Theorem 2. Algorithm CONVERT is 2(1+γ )-approximation for1|r j,qpsd|

∑C j .

Proof. Denote by∑n

j=1 C j(OPT) the optimal objectivevalue for problem 1|qpsd, r j|∑ C j . Since an optimal non-preemptive schedule is a candidate for an optimal pre-emptive schedule, the total completion time of the formerschedule is not less than that of the latter schedule. That is,∑n

j=1 C j(OPT) is not less than∑n

j=1 C j(P ) which is theoptimal objective value for the preemptive schedule P .Together with Corollary 1,∑n

j=1 C j(N) � 2(1 + γ )∑n

j=1 C j(P )

� 2(1 + γ )∑n

j=1 C j(OPT).

The theorem follows. �

838 M. Liu et al. / Information Processing Letters 112 (2012) 835–838

5. Conclusions

This paper investigates the problem of single machinescheduling with psd delivery times and release times. Westudy both preemptive and non-preemptive models of theproblem. Either an optimal algorithm or an approximationalgorithm is proposed for the two models respectively.

Acknowledgements

This research was supported by the National ScienceFoundation of China under Grants 71101106, 71172189,71171149, 70832005 and 71090404/71090400, and theFundamental Research Funds for the Central Universities(2010KJ035).

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