Information Processing Letters 102 (2007) 22–26

www.elsevier.com/locate/ipl

Single machine scheduling with past-sequence-dependent setuptimes and learning effects

Wen-Hung Kuo, Dar-Li Yang ∗

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

Received 7 May 2006; received in revised form 2 August 2006

Available online 28 November 2006

Communicated by W.-L. Hsu

Abstract

This paper studies a single machine scheduling problem with setup times and learning considerations. The setup times areproportional to the length of the already scheduled jobs. That is, the setup times are past-sequence-dependent. It is assumed thatthe learning process reflects a decrease in the process time as a function of the number of repetitions, i.e., as a function of thejob position in the sequence. The following objectives are considered: the makespan, the total completion time, the total absolutedifferences in completion times and the sum of earliness, tardiness and common due-date penalty. Polynomial time algorithms areproposed to optimally solve the above objective functions.© 2006 Elsevier B.V. All rights reserved.

Keywords: Scheduling; Single-machine; Setup times; Learning effect

1. Introduction

In classical scheduling problems, the setup times areconsidered either sequence independent or sequence de-pendent [1]. In the first case, the setup times are usuallyadded to the job processing times while in the secondcase the setup times depend not only on the job currentlybeing scheduled but also on the last scheduled job.Koulamas and Kyparisis [9] first introduced a schedul-ing problem with past-sequence-dependent (p-s-d, forshort) setup times. In this problem, the setup time is de-pendent on all already scheduled jobs. They showed thatthe standard single machine scheduling with p-s-d setup

* Corresponding author. Tel.: +886 5 631-5100; fax: +886 5 6327291.

E-mail address: dlyang@nfu.edu.tw (D.-L. Yang).

0020-0190/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.ipl.2006.11.002

times can be solvable in polynomial time when the ob-jectives are the makespan, the total completion time andthe total absolute differences in completion times, re-spectively. Some extensions with nonlinear p-s-d setuptimes are also considered.

Recently, there is a growing interest in the litera-ture to study scheduling problems with a learning ef-fect. Biskup [3] was the first to analyze the learn-ing effect in single machine scheduling problems. Heshowed that single machine scheduling problems witha learning effect still remain polynomially solvable ifthe objective is to minimize the deviation from a com-mon due date or to minimize the sum of flow times.Later, Mosheiov [14] applied similar solution tech-niques to several other single machine problems. Leeet al. [12] considered the learning effect in a bi-criterionsingle machine scheduling problem. The objective is

W.H. Kuo, D.-L. Yang / Information Processing Letters 102 (2007) 22–26 23

to minimize the linear combination of total comple-tion time and maximum tardiness. They created somedominance properties and applied them to enhance theperformance of a proposed algorithm. Mosheiov andSidney [15] considered a more general learning effectmodel in which the learning effects of some jobs arebetter than those of others in a sequence, i.e., the learn-ing effects are job-dependent. They showed that somescheduling problems with the job-dependent learningeffect remain polynomially solvable. Later, Mosheiovand Sidney [16] provided a polynomial time solutionfor the single-machine scheduling problem to minimizethe number of tardy jobs with general nonincreasingjob-dependent learning curves and common due-date.Lin [13] further showed that the problem with job-dependent due-dates is strongly NP-hard and that theproblem remains NP-hard even when there are only twodue-dates. On the other hand, Cheng and Wang [4] in-troduced a volume-dependent processing time functionto model the learning effects on job processing times ina single machine scheduling problem. The objective isto minimize the maximum lateness. They showed thatthe problem is NP-hard and identified two special caseswhich are polynomially solvable. A survey on this kindof the scheduling research could be found in Bachmanand Janiak [2] and Cheng et al. [5].

The concepts of separated setup time and learning ef-fect have been extensively studied independently in theliterature. However, the scheduling problem simultane-ously with the effects of setup and learning has not beenstudied yet. Therefore, in this paper, we study a singlemachine scheduling problem simultaneously with past-sequence-dependent setup times and a learning effect.The remaining part of the paper is organized as follows.In Section 2, we formulate the model. In Section 3, sev-eral single machine scheduling problems are consideredand shown to be solvable in polynomial time. In Sec-tion 4, all of the scheduling problems are extended tothe problems with the job-dependent learning effect andare proved to remain polynomially solvable. The con-clusions are given in the last section.

2. Problem formulation

To model the effect of p-s-d setup times, we followKoulamas and Kyparisis [9] by assuming that the setuptime of a job is proportional to the length of the alreadyscheduled jobs. The learning effect is modeled in itspopular form of the log-linear curve (see Biskup [3]). Inorder to investigate the effects of p-s-d setup and learn-ing simultaneously, we combine the above models to

constitute our model. The problem is formally describedas follows:

There are n jobs to be processed on a single machine.All jobs are non-preemptive and available for process-ing at time zero. Let pr denote the normal processingtime of job r (Jr , r = 1,2, . . . , n); also, let J[r] and p[r]denote the job occupying position r in a particular se-quence and its normal processing time, respectively. Asin Biskup, it is assumed that the actual processing timeof J[r] when scheduled in position r is given as follows.

pA[r] = p[r]ra, r = 1,2, . . . , n, (1)

where a � 0 is a constant learning index. Also, as inKoulamas and Kyparisis [9], it is assumed that the p-s-dsetup time of J[r] when scheduled in position r is givenas follows.

s[1] = 0 and s[r] = b

r−1∑j=1

pA[j ], r = 2, . . . , n, (2)

where b � 0 is a normalizing constant. For convenience,we denote the learning effect given in Eq. (1) by LEand denote the p-s-d setup given in Eq. (2) by spsd. Inaddition, let Cr denote the completion time of job r

in a sequence. Let d denote a common due date andTr and Er denote the tardiness of job r and the earli-ness of job r , respectively, where Tr = max{Cr − d,0},Er = max{d − Cr,0}. In this paper we will considerthe minimization of the following objective functions:the makespan Cmax = maxr{Cr}, the total completiontime TC = ∑n

r=1 Cr , the total absolute differences incompletion times TADC = ∑n

i=1∑n

j=i |Ci − Cj | andthe sum of earliness, tardiness and common due-datepenalty ETCP = ∑n

r=1(αEr + βTr + γ d) where α,β

and γ are the unit earliness, tardiness and due datepenalty, respectively. Thus, using the three-field nota-tion introduced by Graham et al. [6], the correspondingscheduling problems are denoted by

1|LE, spsd|Cmax, 1|LE, spsd|TC,

1|LE, spsd|TADC and 1|LE, spsd|ETCP,

respectively.

3. Preliminary results

First, a useful lemma is given as follows.

Lemma 1. Let there be two sequences of numbers xi

and yi . In addition, the two sequences are of the samelength. The sum

∑i xiyi of products of the correspond-

ing elements is the least if the sequences are monotonicin the opposite sense.

24 W.H. Kuo, D.-L. Yang / Information Processing Letters 102 (2007) 22–26

Proof. See p. 261 in Hardy et al. [7]. �3.1. The 1|LE, spsd|Cmax scheduling problem

We first derive the makespan of the problem. Clearly,

Cmax =n∑

r=1

(s[r] + pA[r]

) =n∑

r=1

[(n − r)b + 1

]rap[r].

(3)

Eq. (3) can be viewed as the scalar product of twovectors, the wr = [(n − r)b + 1]ra and p[r] vectors,respectively (r = 1, . . . , n). Once the elements of thewr vector have been sorted in strictly decreasing or-der, from Lemma 1, the elements of the p[r] vectorshould be accordingly sorted in strictly increasing order.Based on the above analysis, the optimal sequence forthe 1|LE, spsd|Cmax problem is the well-known shortestprocessing time (SPT) first sequence.

3.2. The 1|LE, spsd|TC scheduling problem

We derive the total completion time of the problem.Clearly,

TC =n∑

r=1

(n − r + 1)(s[r] + pA[r]

)

=n∑

r=1

[(n − r + 1) + b(n − r)(n − r + 1)

2

]rap[r]

=n∑

r=1

(n − r + 1)

(1 + b(n − r)

2

)rap[r]. (4)

Eq. (4) can be viewed as the scalar product of twovectors, the wr = (n − r + 1)(1 + b(n − r)/2)ra andp[r] vectors respectively (r = 1, . . . , n). Since the el-ements of the wr vector are already sorted in strictlydecreasing order, from Lemma 1, the elements of thep[r] vector should be sorted in strictly increasing order.Based on the above analysis, the optimal sequence forthe 1|LE, spsd|TC problem is still the well-known short-est processing time (SPT) sequence.

3.3. The 1|LE, spsd|TADC scheduling problem

We consider a scheduling problem with the objec-tive of minimizing the total absolute variation in thejob completion times (TADC). This scheduling measurewas first considered by Kanet [8]. The TADC of the1|LE, spsd|TADC scheduling problem can be calculatedas follows:

TADC =n∑

i=1

n∑j=i

|Ci − Cj |

=n∑

r=1

(r − 1)(n − r + 1)(s[r] + pA[r]

)

=n∑

r=1

[(r − 1)(n − r + 1)

+ b

n∑l=r+1

(l − 1)(n − l + 1)

]rap[r]. (5)

Eq. (5) can be viewed as the scalar product of twovectors, the

wr =[(r − 1)(n − r + 1)

+ b

n∑l=r+1

(l − 1)(n − l + 1)

]ra (6)

and p[r] vectors respectively (r = 1, . . . , n). Based onthe above analysis and Lemma 1, the optimal sequencefor the 1|LE, spsd|TADC problem can be obtained inO(n logn) time by arranging the elements of the wr andp[r] vectors in opposite orders.

3.4. The 1|LE, spsd|ETCP scheduling problem

This problem is divided into two parts. Firstly theunrestricted common due date problem with earliness,tardiness and due date penalties is introduced. Then apolynomial-time algorithm is given. The goal of the un-restricted common due date problem is to jointly min-imize the weighted earliness, tardiness and due datepenalty. An unrestricted common due date d is a de-cision variable whose value is to be determined. Ifthere are no p-s-d setup and learning effects (i.e., a =b = 0), the 1|LE, spsd|ETCP problem reduces to the1//

∑(αEi +βTi +γ d) problem. Panwalkar et al. [17]

provided some useful results of the 1//∑

(αEi +βTi +γ d) problem as follows:

Theorem 1. For the 1//∑

(αEi + βTi + γ d) problem,

(1) it is optimal to assign the due date at the completiontime of the kth job, where k is the smallest integergreater than or equal to (nβ − nγ )/(α + β);

(2) the optimal schedule is V-shaped, i.e., early jobs arearranged in nonincreasing order of their processingtimes and tardy jobs are arranged in nondecreasingorder of their processing times;

W.H. Kuo, D.-L. Yang / Information Processing Letters 102 (2007) 22–26 25

(3) the positional weight of a job when scheduled inposition r in the sequence is given by

vr = min{nγ + (r − 1)α, (n + 1 − r)β

}.

According to Theorem 1, ETCP = ∑ni=1(αEi +

βTi + γ d) can be rewritten as

ETCP =n∑

i=1

(αE[i] + βT[i] + γ d) =n∑

r=1

vr

(s[r] + pA[r]

)

=n∑

r=1

(vr + b

n−1∑l=r

vl+1

)rap[r]

=n∑

r=1

wrp[r], (7)

where wr = (vr + b∑n−1

l=r vl+1)ra .

Based on the above analysis and Lemma 1, the fol-lowing O(n logn) algorithm is provided to solve the1|LE, spsd|ETCP problem.

Algorithm 1.

Step 1. Assign the optimal due-date at the completiontime of the kth job, where k is the smallest integergreater than or equal to (nβ − nγ )/(α + β), that is,

k =⌈

nβ − nγ

α + β

⌉.

Step 2. Calculate each value of wr, r = 1, . . . , n.Step 3. Assign the job with the longest normal process-

ing time to the position with the smallest value ofwr , the job with the second longer normal process-ing time to the position with the second smallervalue of wr , etc.

The following corollary follows directly from Algo-rithm 1 and Theorem 1.

Corollary 1. For the 1|LE, spsd|ETCP problem,

(1) if b � 1, then there exists an optimal schedule inwhich the jobs are arranged in nondecreasing orderof their processing times;

(2) if b = 0, then the common due date problem reducesto the problem tackled by Mosheiov [14]. In addi-tion, it could be solved by Algorithm 1 of which thecomplexity is O(n logn);

(3) if a = b = 0, then there exists an optimal schedulein which early jobs are arranged in nonincreasingorder of their processing times and tardy jobs arearranged in nondecreasing order of their process-ing times.

Proof. We provide a brief proof as follows.(1) If b � 1, thenw1 > w2 > · · · > wn. Hence, the

result follows from Lemma 1.(2) If b = 0, the problem reduces to that studied by

Mosheiov [14]. Mosheiov solved the problem by formu-lating it as an assignment problem. The complexity ofsolving an assignment problem is O(n3). In fact, the for-mulation of an assignment problem is not necessary, theparticular problem (b = 0) could be solved by definingthe weights wr = vrr

a first, and then applying a simplesorting procedure as Algorithm 1.

(3) If a = b = 0, then the 1|LE, spsd|ETCP problemreduces to the 1||∑(αEr +βTr + γ d) problem. There-fore, the result follows directly from Theorem 1. �4. Extensions

In this section, all of the scheduling problems areextended by the introduction of job-dependent learningeffects and they are shown that they are proved to re-main polynomially solvable.

Under the job-dependent learning environment, allof the jobs have different learning rates ai � 0. As inBiskup, let xir be a 0/1 variable such that xir = 1 if Ji isthe r th job to be processed and xir = 0 otherwise. Thenall of the scheduling problems in Section 3 can be for-mulated as the following assignment problems:

minn∑

i=1

n∑r=1

Wirpixir

s.t.n∑

i=1

xir = 1, r = 1,2, . . . , n,

n∑r=1

xir = 1, i = 1,2, . . . , n,

xir = 0 or 1, i, r = 1,2, . . . , n,

where

Wir = [(n − r)b + 1

]rai

for the makespan problem,

Wir = (n − r + 1)

(1 + b(n − r)

2

)rai

for total completion time problem,

Wir =[(r − 1)(n − r + 1)

+ b

n∑l=r+1

(l − 1)(n − l + 1)

]rai

for the total absolute differences in completion timesproblem, and

26 W.H. Kuo, D.-L. Yang / Information Processing Letters 102 (2007) 22–26

Wir =(

vr + b

n−1∑l=r

vl+1

)rai

for the sum of earliness, tardiness and common due-

date penalty problem.

5. Conclusions

This paper considers a single machine schedulingproblem with setup times and learning considerations.The setup times are proportional to the length of thealready scheduled jobs. The learning effect is also in-vestigated in the scheduling environments, it is assumeda learning process reflects a decrease in the process timeas a function of the number of repetitions. The follow-ing objective functions are considered: the makespan,the total completion time, the total absolute differencesin completion times and the sum of earliness, tardinessand common due-date penalty. The polynomial time al-gorithms are proposed to optimally solve the above ob-jective functions.

In our study, the job-independent/job-dependentlearning effect of a job is assumed. However, in someother situations, a learning effect may be time-dependent[10,11]. Therefore, it is worthwhile for future researchto investigate p-s-d-setup scheduling problems with dif-ferent learning effects. In these cases, it may be nec-essary to resort to heuristic algorithms for obtainingoptimal sequences.

Acknowledgements

The authors would like to thank two anonymous ref-erees for their helpful comments and suggestions on anearly version of this paper. This research is supported inpart by the National Science Council of Taiwan, Repub-lic of China, under grant number NSC-94-2213-E-150-016.

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