Computers & Industrial Engineering 57 (2009) 762–765

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Computers & Industrial Engineering

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

A single-machine scheduling problem with learning effects in intermittentbatch production q

Dar-Li Yang, Wen-Hung Kuo *

Department of Information Management, National Formosa University, Yun-Lin 632, Taiwan, ROC

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

Article history:Received 17 March 2008Received in revised form 22 November 2008Accepted 6 February 2009Available online 14 February 2009

Keywords:Single-machineSchedulingIntermittentLearningMakespan

0360-8352/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.cie.2009.02.003

q This manuscript was processed by Area Edtior Ma* Corresponding author. Tel.: +886 5 631 5733; fax

E-mail address: whkuo@nfu.edu.tw (W.-H. Kuo).

This paper studies a single-machine scheduling problem with three models of learning and forgettingeffects in intermittent batch production. They are the models of no transmission, partial transmissionand total transmission of learning from batch to batch. The phenomena exist in many realistic productionsystems. The objective is to minimize the makespan. We show that the problems with the models of notransmission and partial transmission of learning from batch to batch are polynomially solvable. We alsoprovide two polynomial time algorithms for two special cases in the problem with the total transmissionmodel.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

In classical scheduling problems, processing times of jobs areassumed to be constant. However, in many realistic situations, be-cause the firms and employees perform the same task repeatedly,they learn how to perform them more efficiently. Therefore, the ac-tual processing time of a job is shorter when it is scheduled later,than earlier in the sequence. This phenomenon is known as the‘‘learning effect” in the literature.

The impact of learning on productivity in manufacturing wasfirst found by Wright (1936). However, Biskup (1999) was the firstto introduce learning effect into scheduling problems. He proposeda learning effect model in which the processing time of a job is afunction of the job position in a sequence. He showed that sin-gle-machine scheduling problems with a learning effect still re-main polynomially solvable if the objective is to minimize thedeviation from a common due date or to minimize the sum of flowtimes. Mosheiov (2001a) provided a polynomial time solution forthe single-machine makespan minimization problem and solvedtwo multi-criteria problems which can be formulated as assign-ment problems. He also showed that the SPT (the shortest process-ing time first) rule does not remain optimal for the minimumflow-time problem on parallel identical machines. Mosheiov(2001b) further showed that the flow-time minimization problem

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with the learning effect on parallel identical machines has a poly-nomial time solution. Mosheiov and Sidney (2003) extended learn-ing effect to be job-dependent, that is, learning rates are differentfrom job to job. They showed that the problems of makespan andtotal flow-time minimization on a single machine, a due-dateassignment problem and total flow-time minimization on unre-lated parallel machines remain polynomially solvable. For more re-search results on scheduling problems with other learning effectmodels under different machine environments, the readers are re-ferred to the review papers of Alidaee and Womer (1990), Biskup(2008) and Cheng, Ding, and Lin (2004).

The above learning effect models tell us that, as cumulative jobsincrease, the processing time of the subsequent job decreases in acontinuous production system. It gives the implication of continu-ous production rather than intermittent batch production. How-ever, many realistic production systems are intermittent, forexample, production systems with the applications of ‘‘group tech-nology”. That is, different products with similar designs and/orproduction processes are grouped together to produce in a produc-tion run. In such intermittent production, it is reasonable to as-sume that if plenty of time has elapsed between production runs,the learning effect would not continue to follow what it was leftwhen production resumes, but that the processing time of the sub-sequent job would revert to a higher level. This suggests a phe-nomenon of forgetting between production runs. Keachie andFontana (1966) also indicated that transmission of learning fromperiod to period can depend on many variables, such as the typeof work performed, the time between manufacturing periods, the

D.-L. Yang, W.-H. Kuo / Computers & Industrial Engineering 57 (2009) 762–765 763

length of time required to manufacture the lot and the labor turn-over. Therefore, there may be total, partial or even no transmissionof learning from one production run to another one. In this paper,we study a scheduling problem with three models of learning andforgetting effects in intermittent batch production.

2. Notations and assumptions

As mentioned above, we consider a single-machine schedulingproblem with learning and forgetting effects in intermittent batchproduction. The problem is developed by using the followingnotations.

m the number of batches (m P 2).Bi the ith batch, i = 1,2, . . .,mni the number of jobs in batch Bi, i = 1,2, . . .,mn the total number of jobs (i.e. n1 + n2 + � � � + nm = n).Jij the jth job in batch Bi, j = 1,2, . . .,ni.ai the learning factor of jobs within batch Bi (ai 6 0).bi the learning factor of batch Bi (bi 6 0).pij the normal processing time of Jij in the original sequence.pr

ij the actual processing time of Jij which is scheduled in therth position in a sequence in batch Bi

pi[k] the normal processing time of Ji[k]which is scheduled in thekth position in a sequence in batch Bi

pki½k� the actual processing time of Ji[k]which is scheduled in the

kth position in a sequence in batch Bi

Cij the completion time of Jij.Ci[k] the completion time of Ji[k]which is scheduled in the kth

position in a sequence in batch Bi.Cmax the makespan of all jobs.

There are n jobs grouped into m batches and processed on a sin-gle machine. All jobs are available at time zero. The normal pro-cessing time of a job is incurred when the job is scheduled firstin the first production batch. The actual processing times of thesubsequent jobs are smaller than their normal processing times be-cause of the learning effect. Assume that the actual processing timeof a job is a decreasing function of its position in a sequence. Usu-ally, the learning effect can be accumulated through completingjobs. However, if plenty of time has elapsed between productionruns, it may incur a forgetting effect. That is, there may be total,partial or even no transmission of learning from batch to batch.A single-machine scheduling problem with three models of learn-ing and forgetting effects is studied in the following sections.

3. Model I: no transmission of learning from batch to batch

In the first model, we consider that there is no transmission oflearning from batch to batch. The objective of the single-machinescheduling problem is to minimize the makespan of all jobs. Asmentioned in Biskup (1999), we assume that the actual processingtime of job Jij when scheduled in position r of batch Bi, is given by

prij ¼ pijr

ai :

Hence, the makespan of all jobs is as follows:

Cmax ¼Xm

i¼1

Xni

j¼1

pi½j�jai :

For convenience, let LE denote the learning effect, B denote thatthe problem is an intermittent batch production problem and Tno

denote that there is no transmission of learning from batch tobatch. Therefore, following the three-field notation of Graham,Lawler, Lenstra, and Rinnooy Kan (1979), the proposed problemis denoted by 1/B, LE, Tno/Cmax.

Theorem 1. For the problem of 1/B, LE, Tno/Cmax, there exists anoptimal schedule that satisfies the following conditions: (a) the jobswithin a batch are sequenced in non-decreasing order of theirnormal processing times and (b) the batches can be sequenced inany order.

Proof. The problem of arranging the job sequence optimallywithin a batch is the same as the problem of 1/LE/Cmax. Mosheiov(2001a) proved that the optimal schedule is to sequence the jobsin non-decreasing order of their normal processing times. There-fore, the theorem follows because there is no transmission of learn-ing from batch to batch. h

The optimal job sequence within a certain batch Bi can be ob-tained by a sorting algorithm and thus taking O(ni logni) time.Hence, the total running time to sequence jobs of all batches isPm

i¼1Oðni log niÞ. On the other hand, the running time to sequencethe batches in any order is O(1). Therefore, the overall complexityof the problem of 1/B, LE, Tno/Cmax is O(n logn) (see Kuo & Yang,2006a).

4. Model II: partial transmission of learning from batch to batch

In the second model, we consider partial transmission of learn-ing from batch to batch in the single-machine scheduling problem.We assume that the learning effect of jobs within a batch is thesame as that in the first model. In addition, the actual processingtime of batch Bi when scheduled in the rth batch is defined asfollows:

Pir ¼ Pirbi

where Pi is the total processing time of jobs within batch Bi if thereis no transmission of learning from batch to batch. That is,Pi ¼

Pnik¼1pi½k�k

ai .Then, the makespan of all jobs is calculated as follows:

Cmax ¼Xm

i¼1

ðpi½1�1ai þ pi½2�2

ai þ � � � þ pi½ni �naii Þi

bi ¼Xm

i¼1

Piibi ð1Þ

Let Tpart denote that there is partial transmission of learningfrom batch to batch. The proposed problem is denoted by 1/B, LE,Tpart/Cmax. As in Biskup (1999), let xir be a 0/1 variable such thatxir = 1 if batch Bi is the rth batch to be processed and xir = 0 other-wise. Then the problem of 1/B, LE, Tpart/Cmax can be formulated asthe following assignment problem:

minXm

i¼1

Xm

r¼1

Pirbi xir ð2Þ

s:t:Xm

i¼1

xir ¼ 1; r ¼ 1;2; . . . ;m; ð3Þ

Xm

r¼1

xir ¼ 1; i ¼ 1;2; . . . ;m; ð4Þ

xir ¼ 0 or 1; i; r ¼ 1;2; . . . ;m: ð5Þ

Since Pi of batch Bi is not affected by the batch sequence, fromTheorem 1, Pi can be minimized by sequencing the correspondingjobs in non-decreasing order of their normal processing times.Based on the above analysis, a simple algorithm to determine theoptimal schedule for the problem of 1/B, LE, Tpart/Cmax is developedas follow.

Algorithm 1

Step 1: Arrange jobs within each batch in non-decreasing orderof their normal processing times.

764 D.-L. Yang, W.-H. Kuo / Computers & Industrial Engineering 57 (2009) 762–765

Step 2: Formulate the corresponding assignment problem as Eq.(2) and determine the batch sequence according to the solutionof the corresponding problem.

From the analysis in Section 3, the complexity of Step 1 is O(n

logn). On the other hand, Step 2 is to solve an assignment problemand thus the complexity of Step 2 is O(m3). Thus, the overall com-plexity of Algorithm 1 is O(n logn + m3).

Corollary 1. If the learning factors of all batches are equal, i.e. bi = b,then for the problem of 1/B, LE, Tpart/Cmax, there exists an optimalschedule that satisfies the following conditions: (a) the jobs within abatch are sequenced in non-decreasing order of their normal process-ing times and (b) the batches are sequenced in non-decreasing order ofPi.

Proof. First, the proof of part (a) is the same as that in Theorem 1.Next, because bi = b, to minimize

Pmi¼1Pii

b is equivalent tominimizing the makespan of the problem of 1/LE/Cmax if batchesare taken as jobs. Then, the result in part (b) of Corollary 1follows. h

From Corollary 1, if bi = b, the complexity of the problem of 1/B,LE, Tpart /Cmax is reduced to O(n logn).

5. Model III: total transmission of learning from batch to batch

In the third model, we consider total transmission of learningfrom batch to batch in the single-machine scheduling problem.Without loss of generality, assume that batch Bi is sequenced inthe ith batch. Then, the actual processing time of job Jij whenscheduled in position r in batch Bi is as follows:

prij ¼ pij r þ

Xi�1

k¼1

nk

!ai

:

Hence, the makespan of all jobs is calculated as follows:

Cmax ¼Xm

i¼1

Xni

r¼1

pi½r� r þXi�1

k¼1

nk

!ai

: ð6Þ

Let Ttotal denote total transmission of learning from batch to batch.Then the proposed problem is denoted by 1/B, LE, Ttotal/Cmax.

Theorem 2. For any batch sequence of the 1/B, LE, Ttotal/Cmax problem,the total processing time of jobs within the batch is minimized bysequencing jobs in non-decreasing order of their normal processingtimes.

Proof. The theorem can be easily proved by using simple job inter-changing technique. h

Corollary 2. For the problem of 1/B, LE, Ttotal/Cmax, there exists anoptimal schedule by sequencing jobs within each batch in non-decreasing order of their normal processing times.

Proof. The result follows directly from Theorem 2. h

In the following, two special cases of the problem of 1/B, LE,Ttotal/Cmax are discussed in Theorem 3 and Algorithm 2, res-pectively.

Definition 1. Bi is dominated by Bj, or Bj dominates Bi iffmaxfpikjk ¼ 1;2; :::;nig 6minfpjkjk ¼ 1;2; :::;njg. The symbolBi � Bj denotes that Bi is dominated by Bj.

Definition 2. The batches form an increasing sequence of domi-nating batches iff B1 � B2 � � � � � Bm.

Theorem 3. If B1 � B2 � � � � � Bm and a1 = a2 = � � � = am, then for theproblem of 1/B, LE, Ttotal/Cmax, there exists an optimal schedule satisfiesthe following conditions:

(a) the jobs within a batch are sequenced in non-decreasing

order of their normal processing times.

(b) the batches are arranged as an increasing sequence of dom-inating batches.

Proof. The theorem can be easily proved by using simple jobinterchanging technique. h

Next, if the job numbers of all batches are equal (i.e.n1 ¼ n2 ¼ � � � ¼ nm ¼ n=m ¼ �n), then the proposed problem can beformulated as an assignment problem. Again, let xir be a 0/1 variablesuch that xir = 1 if batch Bi is the rth batch to be processed and xir = 0otherwise. Then, the problem to minimize the makespan of all jobsis formulated as follows.

minXm

i¼1

Xm

r¼1

X�n

j¼1

pi½j�ððr � 1Þ�nþ jÞai xir ð7Þ

s:t:Xm

i¼1

xir ¼ 1; r ¼ 1;2; . . . ;m; ð8Þ

Xm

r¼1

xir ¼ 1; i ¼ 1;2; . . . ;m; ð9Þ

xir ¼ 0 or 1; i; r ¼ 1;2; . . . ;m: ð10Þ

Based on Theorem 2 and the above analysis, a simple algorithmto determine the optimal schedule for the problem of 1/B, LE, Ttotal/Cmax is developed as follow.

Algorithm 2

Step 1: Arrange jobs within each batch in non-decreasing orderof their normal processing times.Step 2: Formulate the corresponding assignment problem as Eq.(3) and determine the batch sequence according to the solutionof the corresponding problem.

Note that Step 1 can be obtained by a sorting algorithm andthus it takes Oð�n log �nÞ time. Step 2 is to solve an assignment prob-lem and thus it takes O(m3) time. Thus, the overall time complexityof Algorithm 2 is Oð�n log �nþm3Þ.

6. Conclusions

This paper studies a single-machine scheduling problem withthree models of learning and forgetting effects in intermittentbatch production. They are the models of no transmission, partialtransmission and total transmission of learning from batch tobatch, respectively. The objective is to minimize the makespan.We show that the problem with the models of no transmissionand partial transmission of learning from batch to batch is polyno-mially solvable. We also provide two polynomial time algorithmsto find the optimal solutions of two special cases in the problemwith the model of total transmission of learning from batch tobatch. However, the complexity of the problem with the modelof total transmission of learning from batch to batch is an openquestion. Therefore, it is an interesting topic for the future re-search. Besides, in this study, the learning effect of a job dependson its position in a schedule. There are other learning effect modelsstudied in the literature. Thus, it is also worthwhile to consider an-other learning effect model in intermittent batch production, forexample, a time-dependent learning effect model proposed byKuo and Yang (2006a, 2006b).

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Acknowledgements

The authors are grateful to the referees for their constructivecomments on an earlier version of this paper. This research is sup-ported in part by the National Science Council of Taiwan, Republicof China, under Grant Number NSC-96-2221-E-150-060.

References

Alidaee, B., & Womer, N. K. (1990). Scheduling with time dependent processingtimes: Review and extensions. Journal of the Operational Research Society, 50,711–729.

Biskup, D. (1999). Single-machine scheduling with learning considerations.European Journal of Operational Research, 115, 173–178.

Biskup, D. (2008). A state-of-the-art review on scheduling with learning effects.European Journal of Operational Research, 188, 315–329.

Cheng, T. C. E., Ding, Q., & Lin, B. M. T. (2004). A concise survey of scheduling withtime-dependent processing times. European Journal of Operational Research, 152,1–13.

Graham, R. L., Lawler, E. L., Lenstra, J. K., & Rinnooy Kan, A. H. G. (1979).Optimization and approximation in deterministic sequencing and scheduling: Asurvey. Annals Discrete Mathematics, 5, 287–326.

Keachie, E. C., & Fontana, R. J. (1966). Effects of learning on optimal lot size.Management Science, 13(2), B102–B108.

Kuo, W. H., & Yang, D. L. (2006a). Single-machine group scheduling with a time-dependent learning effect. Computers & Operations Research, 33(8), 2099–2112.

Kuo, W. H., & Yang, D. L. (2006b). Minimizing the total completion time in a singlemachine scheduling problem with a time-dependent learning effect. EuropeanJournal of Operational Research, 174(2), 1184–1190.

Mosheiov, G. (2001a). Scheduling problems with learning effect. European Journal ofOperational Research, 132, 687–693.

Mosheiov, G. (2001b). Parallel machine scheduling with learning effect. Journal ofthe Operational Research Society, 52, 1–5.

Mosheiov, G., & Sidney, J. B. (2003). Scheduling with general job-dependent learningcurves. European Journal of Operational Research, 147, 665–670.

Wright, T. P. (1936). Factors affecting the cost of airplanes. Journal of AeronauticalScience, 3, 122–128.