Int. J. Production Economics 144 (2013) 418–421

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Int. J. Production Economics

0925-52http://d

n CorrE-m

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

Scheduling deteriorating jobs with past-sequence-dependentdelivery times

Ming Liu a, Shijin Wang a,n, Chengbin Chu a,b

a School of Economics & Management, Tongji University, Shanghai 200092, PR Chinab 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

Article history:Received 5 December 2011Accepted 25 February 2013Available online 17 March 2013

Keywords:SchedulingPast-sequence-dependent delivery timesDeteriorating jobs

73/$ - see front matter & 2013 Elsevier B.V. Ax.doi.org/10.1016/j.ijpe.2013.03.009

esponding author. Tel.: þ86 150 2661 3178.ail address: shijinwang0223@yahoo.com.cn (S

a b s t r a c t

In this note, we study parallel machine scheduling problem with past-sequence-dependent deliverytimes and a deterioration effect. We present polynomial algorithms for the problem with the totalworkload, the total completion time, the total absolute differences in completion times objectives.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

The situations in which the processing time is time-dependent areoften encountered in real-life applications. Examples can be found inseveral circumstances, such as steel production, machine maintenanceand fire fighting, where any delay in dealing with a task may increaseits processing time. Such problems are generally known as schedulingwith deterioration effect. The concept of deterioration effect was firstintroduced in single machine scheduling by Gupta and Gupta (1988)and Browne and Yechiali (1990). They defined linear deteriorating jobwhose actual processing time increases in its starting time. Moreprecisely, the actual processing time of job Jj is pj ¼ ajþbjt (bj40),where aj is the basic or normal processing time, bj the deteriorationrate, and t the starting time. The reader is referred to Gawiejnowicz(2008) for further information on time-dependent scheduling.

In addition to jobs' deterioration themselves, manufacturingenvironment may also have a great influence on their processingtimes. Recent empirical studies in several industries have demon-strated that the manufacturing environment (or waiting time inthe hostile environment) has an adverse effect on the totalprocessing time of a job. For example, an electronic componentmay be exposed to certain electromagnetic fields while waiting inthe machine's pre-processing area and regulatory authoritiesrequire the component to be “treated”. The waiting time-inducedadverse effect usually does not impede the job's suitability to beprocessed and it can be eliminated after the main processing of thejob with an extra time. Such an extra time for eliminating thewaiting time-induced adverse effect is viewed as a past-sequence-

ll rights reserved.

. Wang).

dependent (psd) delivery time (Koulamas and Kyparisis, 2010).Notice that since the treatment of the adverse effect for a jobdoes not occupy any machine and has no relation with theschedule of the job's main processing, the treatment is performedimmediately after the component's completion on the machine.

Now, we briefly review related scheduling problems, using three-field notation (Pinedo, 2008) with extensions proposed inGawiejnowicz (2008). Koulamas and Kyparisis (2010) assumed thatthe psd delivery time of a job is proportional to the job's waiting timeand presented several results on single-machine scheduling problem.For the makespan minimization problem, they showed that theproblem 1jqpsdjCmax can be solved in O(n) time by arranging the largestjob at last. Further they showed that 1jqpsdjLmax (the problem tominimize maximum lateness), 1jqpsdjTmax (the problem to minimizemaximum tardiness) and 1jqpsdj∑Uj (the problem to minimize thenumber of tardy jobs) can be solved polynomially (by reduction to theircounterparts in classical scheduling theory Pinedo, 2008). For the samemachine setting, more results have been presented in Liu et al. (2012):(1) SWPT rule is optimal for 1jqpsdj∑wjCj, (2) a modified version ofWeighted Discounted Shortest Processing Time (WDSPT) rule has beenshown to be optimal for 1jqpsdj∑wjð1−e−rCj Þ, (3) an extension of thealgorithm proposed in Panwalkar et al. (1982) has been proved to beoptimal for a common due date problem 1jqpsdj∑ðP1dþP2ErþP3TrÞ,and (4) SPT rule has been shown to be optimal for 1jqpsdj∑wjTj subjectto agreeable due date and agreeable weight.

To the best of our knowledge, there is no result on machinescheduling problems with psd delivery times and deterioration effect.In this note we consider time-dependent counterparts of resultspresented in Koulamas and Kyparisis (2010) and Liu et al. (2012). Theremaining part of this note is organized as follows. In Section 2, weformulate the parallel machine models. In Section 3, for different

M. Liu et al. / Int. J. Production Economics 144 (2013) 418–421 419

objective function, we propose polynomial algorithms. In the lastsection we summarize the contribution of this note.

2. Preliminary results

There are given n jobs and m≥1 identical parallel machines,M1,…,Mm. Each machine can handle one job at a time andpreemption is not allowed. Denote by Ωi the set of jobs assignedto machine Mi, i∈f1,…,mg. Thus, Ωi1∩Ωi2 ¼∅ for i1≠i2, andΩ1∪⋯∪Ωm ¼ fJ1,…,Jng. If m≥n, the problem is trivial. Therefore,we assume that mon throughout the note.

Given a schedule, reindex the jobs on each machine. Let nidenote the number of jobs scheduled on machine Mi for i¼ 1,…,m.Let Ji½j� indicate the job scheduled on machine Mi in the jthposition. Denote by ai½j� and pi½j� (i∈f1,…,mg and j∈f1,…,nig) thenormal processing time and actual processing time of job Ji½j�,respectively. Denote by Si½j� the starting time of job Ji½j�. In theenvironment with psd delivery times, the processing of Ji½j� must befollowed immediately by its psd delivery time qi½j�.

We assume that the actual processing time of job Ji½j� (i¼ 1,…,mand j¼ 1,…,ni) is given by

pi½j� ¼ ai½j� þbSi½j� ¼ ai½j� þb ∑j−1

k ¼ 1pi½k�, ð1Þ

where b≥0 is a deteriorating rate. Also, as in Koulamas andKyparisis (2010), we assume that the psd delivery time qi½j� isformulated as

qi½j� ¼ γSi½j� ¼ γ ∑j−1

k ¼ 1pi½k�, j¼ 1,…,ni: ð2Þ

Let Ci½j� denote the completion time of job Ji½j� in a schedule (i.e.,the completion time of the processing of Ji½j� on the machine plusthe job's psd delivery time). Therefore,

Ci½j� ¼ Si½j� þpi½j� þqi½j�¼ ð1þbþγÞSi½j� þai½j�

¼ ð1þbþγÞ ∑j−1

k ¼ 1pi½k� þai½j�, j¼ 1,…,n:

Given a schedule, by mathematical induction, it can be verifiedthat

pi½j� ¼ ai½j� þb ∑j−1

k ¼ 1ð1þbÞj−1−kai½k�, ð3Þ

and that

Ci½j� ¼ ð1þγþbÞ ∑j−1

k ¼ 1ð1þbÞj−1−kai½k� þai½j�, j¼ 1,…,ni: ð4Þ

We define ∑0k ¼ 1ai½k�≔0 for i∈f1,…,mg.

First, we introduce a useful lemma.

Lemma 1 (Hardy et al., 1967). Let there be two sequences ofnumbers xi and yi. In addition, the two sequences are of the samelength. The sum ∑ixiyi of products of the corresponding elements isthe least if the sequences are monotonic in the opposite sense.

Let ðn1,…,nmÞ denote a vector such that n1,…,nm are naturalnumbers (of jobs) and n1þ⋯þnm ¼ n. The following lemma givesan upper bound on the number of ðn1,…,nmÞ vectors.

Lemma 2. An upper bound on the number of ðn1,…,nmÞ vectors isequal to Oððnþ1Þm−1Þ.

Proof. For machine M1, n1 is chosen from {0, 1,…,n}. After n1 isdetermined, there are nþ1−n1 options left for n2, i.e., n2 is chosenfrom f0,n−n1gDf0,1,…,ng. For machines M3,…,Mm−1, we similarly

select n3,…,nm−1. Note that for the last machine Mm, nm isdetermined after choosing n1,…,nm−1, because ∑m

i ¼ 1ni ¼ n. Con-sidering that m machines are identical, there are ðnþ1Þm−1=m!possibilities for ðn1,…,nmÞ vector. This completes the proof. □

In fact, there is a slightly tighter upper bound on the number ofðn1,…,nmÞ vectors (see Lemma 3 in Appendix).

3. Parallel machine scheduling problems

In this section, we present our main results on parallel machinescheduling with psd delivery times and deterioration effect.

Let ∑nil ¼ 1∑

nik ¼ ljCil−Cikj denote the total absolute deviation (of

jobs completion times) on machine Mi. Let Cimax indicate the load

of machine Mi, i.e., Cimax ¼maxr ¼ 1,…,ni fCirg. We respectively con-

sider the minimization of the following objective functions: thesum of the total absolute deviation on all machines∑m

i ¼ 1∑nil ¼ 1∑

nik ¼ ljCil−Cikj, the total workloads on all machines

∑mi ¼ 1C

imax and the total completion time ∑m

i ¼ 1∑nij ¼ 1Cij. We will

denote the corresponding scheduling problems by Pmjpj ¼ajþbt,qpsdj∑m

i ¼ 1∑nil ¼ 1∑

nik ¼ ljCil−Cikj, Pmjpj ¼ ajþbt,qpsdj ∑m

i ¼ 1Cimax,

and Pmjpj ¼ ajþbt,qpsdj∑mi ¼ 1∑

nij ¼ 1Cij, respectively.

3.1. Pmjpj ¼ ajþbt,qpsdj∑mi ¼ 1∑

nil ¼ 1∑

nik ¼ ljCil−Cikj problem

In this subsection, we show that Pmjpj ¼ ajþbt,qpsdj∑m

i ¼ 1∑nil ¼ 1∑

nik ¼ ljCil−Cikj can be solved in polynomial time.

This scheduling problem was first considered by Kanet (1981). Ifthe number of jobs on machine Mi is known in advance, theexpression ∑ni

l ¼ 1∑nik ¼ ljCi½l�−Ci½k�j can be computed as follows:

∑ni

r ¼ 1∑ni

j ¼ rjCi½r�−Ci½j�j

¼ ∑ni

r ¼ 1ð2r−1−niÞCi½r�

¼ ∑ni

r ¼ 1ð2r−1−niÞ½ð1þbþγÞ ∑

r−1

k ¼ 1ð1þbÞr−1−kai½k� þai½r��

¼ ð1þγþbÞ ∑ni−1

r ¼ 1∑ni

k ¼ rþ1ð2k−1−niÞð1þbÞk−r−1ai½r�

þ ∑ni

r ¼ 1ð2r−1−niÞai½r�

¼ ∑n

r ¼ 1wirai½r�,

where

wir ¼ð1þγþbÞ ∑

ni

k ¼ rþ1ð2k−1−niÞð1þbÞk−r−1, r¼ 1,…,ni−1

ni−1, r¼ ni:

8><>:

Therefore,

∑m

i ¼ 1∑ni

r ¼ 1∑ni

j ¼ rjCi½r�−Ci½j�j ¼ ∑

m

i ¼ 1∑ni

r ¼ 1wirai½r�:

The above equation can be viewed as the scalar product of thewir vector and ai½r� vector for i¼ 1,…,m and r¼ 1,…,ni. Therefore, ifthe number ni of jobs on machine Mi is known, by Lemma 1, alljobs are sorted in non-decreasing order of their normal processingtimes first. Then the largest job (i.e., the job with largest normalprocessing time) is assigned to the position with the smallest valueof wir, the second largest job to the position with the secondsmallest value of wir, and so on. The time complexity of sortingalgorithm is Oðn log nÞ.

Þ:

M. Liu et al. / Int. J. Production Economics 144 (2013) 418–421420

Let Pðn,mÞ ¼ ðn1,…,nmÞ denote the allocation vector. For a givenPðn,mÞ vector, we know the problem can be solved in Oðn log nÞtime. Together with Lemma 2, we have the following theorem.

Theorem 1. The problem Pmjpj ¼ ajþbt, qpsdj∑mi ¼ 1∑

nir ¼ 1

∑nij ¼ rjCir−Cijj can be solved in Oððnþ1Þm−1n log nÞ time.

3.2. Pmjpj ¼ ajþbt,qpsdj∑Cimax problem

In this subsection, we deal with the problem Pmjpj ¼ ajþbt,qpsdj∑Ci

max. Similar to the above analysis, if ni (i¼ 1,…,m) are fixed inadvance, by (4), the workload on machine Mi can be computed asfollows:

Cimax ¼ ∑

ni

r ¼ 1wirai½r�,

where for i¼ 1,…,m,

wir ¼ð1þbþγÞð1þbÞni−1−k, r¼ 1,…,ni−1;1, r¼ ni:

(

Therefore, the sum of workload on all machines is

∑m

i ¼ 1Cimax ¼ ∑

m

i ¼ 1∑ni

r ¼ 1wirai½r�,

where for i¼ 1,…,m,

wir ¼ð1þbþγÞð1þbÞni−1−k, r¼ 1,…,ni−1;1, r¼ ni:

(

Similarly, the above equation can be viewed as the scalarproduct of the wir vector and pi½r� vector. Therefore, if the numberni of jobs on machine Mi is known, by Lemma 1, all jobs must bescheduled in non-decreasing order of their normal processingtimes first. Then the largest job is assigned to the position with thesmallest value of wir, the second largest job to the position withthe second smallest value of wir, and so on. The sorting operationtakes Oðn log nÞ. Together with Lemma 2, we obtain the followingtheorem.

Theorem 2. The problem Pmjpj ¼ ajþbt, qpsdj∑Cimax can be solved in

Oððnþ1Þm−1n log nÞ time.

3.3. Pmjpj ¼ ajþbt,qpsdj∑mi ¼ 1∑

nij ¼ 1Cij problem

In this subsection, we consider the problem of minimizing thetotal completion time, i.e., Pmjpj ¼ ajþbt,qpsdj∑m

i ¼ 1∑nij ¼ 1Cij. For

machine Mi, by (4), we derive the total completion time on thismachine.

∑ni

r ¼ 1Ci½r� ¼ ∑

ni

r ¼ 1wirai½r�,

where for i¼ 1,…,m,

wir ¼1þð1þbþγÞ½ð1þbÞr−1�

b, r¼ 1,…,ni−1;

1, r¼ ni:

8<:

Therefore, the total completion time is

∑m

i ¼ 1∑ni

r ¼ 1Ci½r� ¼ ∑

m

i ¼ 1∑ni

r ¼ 1wirai½r�,

where for i¼ 1,…,m,

wir ¼1þð1þbþγÞ½ð1þbÞr−1�

b, r¼ 1,…,ni−1;

1, r¼ ni:

8<:

Proceeding in a similar way as previously, we obtain thefollowing result.

Theorem 3. The problem Pmjpj ¼ ajþbt,qpsdj∑mi ¼ 1∑

nij ¼ 1Cij can be

solved in Oððnþ1Þm−1n log nÞ time.

4. Conclusions

This note addressed machine scheduling problem with psddelivery times and deterioration effect. The delivery time of a jobis proportional to its waiting time, i.e., the total processing time ofthe already scheduled jobs. Polynomially solvable problems havebeen explored for parallel-machine setting. Future research mayinclude (1) scheduling deteriorating jobs with psd delivery time onflow shop and open shop, (2) incorporating psd delivery timeswith deteriorating maintenance activity presented in Cheng et al.(2012).

Acknowledgments

We would like to thank the anonymous referee(s) for theirconstructive comments and suggestion. The first author issupported by the Fundamental Research Funds for the CentralUniversities (2010KJ035) and Ph.D. Programs Foundation of Min-istry of Education of China (20110072120014). This research wasalso supported by the National Science Foundation of China underGrants 71101106, 71171149, and 71090404/71090400.

Appendix A

Define a function Cðm,n,pÞ to be the maximum number ofvectors ðn1,…,nmÞ such that n1þn2þ⋯þnm ¼ n andn1≤n2≤⋯≤nm≤p.

Observe that

Cðm,n,pÞ ¼ ∑nm≤p

Cðm−1,n−nm,nmÞ,

and by induction,

Cðm,n,pÞ ¼ ∑nm≤p

∑nm−1≤nm

⋯ ∑n2≤n3

Cð1,n−nm−⋯−n2,n2Þ:

Due to Cð1,n−nm−⋯−n2,n2Þ ¼ 1 (i.e., arrange all n−nm−⋯−n2 jobson the only one machine), it follows:

Cðm,n,pÞ ¼ ∑nm≤p

∑nm−1≤nm

⋯ ∑n2≤n3

1¼ ðp−nmþ1Þðnm−nm−1þ1Þ⋯ðn3−n2þ1

Set p¼n, we obtain

Cðm,n,nÞ ¼ ðn−nmþ1Þðnm−nm−1þ1Þ⋯ðn3−n2þ1Þ:Observe that there are m−1 parenthesizes in the formula aboveand their sum is equal to n−n2þm−1. By the fact that geometricmean is less than the arithmetic mean, i.e.,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðn−nmþ1Þðnm−nm−1þ1Þ⋯ðn3−n2þ1Þm−1

p≤n−n2þm−1

m−1,

we derive

Cðm,n,nÞ≤ n−n2

m−1þ1

� �m−1≤

nm−1

þ1� �m−1

:

Lemma 3. An upper bound on the number of ðn1,…,nmÞ vectors isequal to Oððn=ðm−1Þþ1Þm−1Þ.

References

Browne, S., Yechiali, U., 1990. Scheduling deteriorating jobs on a single processor.Operations Research 38, 495–498.

Cheng, T.C.E., Yang, S.J., Yang, D.L., 2012. Common due-window assignment andscheduling of linear time-dependent deteriorating jobs and a deterioratingmaintenance activity. International Journal of Production Economics 135,154–161.

M. Liu et al. / Int. J. Production Economics 144 (2013) 418–421 421

Gawiejnowicz, S., 2008. Time-Dependent Scheduling. Springer-Verlag, Berlin,Heidelberg.

Gupta, J., Gupta, S., 1988. Single facility scheduling with nonlinear processing times.Computer Industrial Engineering 14, 387–393.

Hardy, G.H., Littlewood, J.E., Polya, G., 1967. Inequalities. Cambridge UniversityPress, London.

Kanet, J.J., 1981. Minimizing variation of flow time in a single machine systems.Management Science 27, 1453–1459.

Koulamas, C., Kyparisis, G.J., 2010. Single-machine scheduling problems with past-sequence-dependent delivery times. International Journal of Production Eco-nomics 126, 264–266.

Liu, M., Zheng, F.F., Chu, C.B., Xu, Y.F., 2012. New results on single-machinescheduling with past-sequence-dependent delivery times. Theoretical Compu-ter Science 438, 55–61.

Panwalkar, S.S., Smith, M.L., Seidmann, A., 1982. Common due date assignment tominimize total penalty for the one machine scheduling problem. OperationsResearch 30, 391–399.

Pinedo, M., 2008. Scheduling: Theory, Algorithms, and Systems, 3rd ed. PrenticeHall, New York.