Applied Mathematics and Computation 226 (2014) 415–422

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Applied Mathematics and Computation

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

Single-machine scheduling with piece-rate maintenanceand interval constrained position-dependent processing times

0096-3003/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.10.034

⇑ Corresponding author.E-mail address: zhangyl@seu.edu.cn (Y.L. Zhang).

Pengfei Xue a, Yulin Zhang a,⇑, Xianyu Yu a,b

a School of Economics and Management, Southeast University, Nanjing 210096, Chinab School of Science, East China Institute of Technology, Fuzhou, Jiangxi 344000, China

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

Keywords:Single-machineSchedulingPosition-dependentMaintenanceTotal cost

This paper investigates single-machine scheduling problems with piece-rate machinemaintenance and interval constrained actual processing time. The actual processing timeof a job is a general function of the normal job processing time and the position in jobsequence, and it is required to restrict in given interval otherwise earliness or tardinesspenalty should be paid. The maintenance duration studied in the paper is a time-depen-dent linear function. The objective is to find jointly the optimal frequency to perform main-tenances and the optimal job sequence to minimize the total cost, which is a linear functionof the makespan, total earliness and total tardiness. There is shown that the problem can beoptimally solved in O(n4) time. There is also shown that two special cases of the problemcan be optimally solved by lower order algorithms.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

Recent developments in the field of job scheduling have triggered into a growing interest towards learning or aging effect.In case of the learning effect, the actual processing time of a job will be shorter when it is scheduled later in a sequence.While in case of the aging effect, the actual processing time of a job will be longer when it is scheduled later in a sequence.Some papers discussed time-dependent scheduling problems. The relevant papers include Bachman et al. [1], Cheng et al. [2]and Ji and Cheng [3]. Mosheiov and Oron [4] and Hsu et al. [5] discussed position-dependent scheduling problems. For sched-uling problems with position-dependent effect, the position-dependent function as specific linear or exponential functionwas considered by Cheng and Wang [6], Bachman and Janiak [7], Wang and Xia [8], Biskup [9], Kuo and Yang [10], Yang, Yangand Cheng [11], Yang and Yang [12] and Zhao and Tang [13]. Some relevant references for the combination of the exponentialfunction and linear function are: Lee [14], Cheng, Wu and Lee [15], Wang [16], Wang and Cheng [17], Cheng and Lee [18], Yinet al. [19] and Lai and Lee [20]. However, all these papers consider learning or aging model with specific processing timefunction. Only a few papers study general position-dependent processing times. Mosheiov and Sidney [21] and Wang andWang [22] considered a scheduling problem with a general position-dependent learning function. Mosheiov [23] studieda scheduling problem with position-dependent job processing times which is not restricted to a monotone function.

In addition, it is well-known that the production efficiency can be improved by performing maintenance in manufactur-ing processes. Some scheduling researchers considered that the machine is maintained exactly once during the planning per-iod, and the duration of maintenance is not fixed. Some recent relevant references are: Kubzin and Strusevich [24], Mosheiovand Sidney [25], Cheng et al. [26], Mor and Mosheiov [27] and Cheng et al. [28]. Some scheduling researchers assumed thatthe duration of each maintenance is given. Liao and Chen [29] and Ji et al. [30] considered single-machine scheduling

416 P. Xue et al. / Applied Mathematics and Computation 226 (2014) 415–422

problems with periodic maintenance activities, where each maintenance activity is required after a periodic time interval.Ji and Cheng [31] and Yang et al. [32] assumed that the machine may have multiple rate-modifying activities over thescheduling horizon.

Another popular topic in recent years is scheduling problem with job completion time due window. To cope with globalcompetition and to improve customer demand, jobs should be finished as close as possible to their due-dates. If a job is fin-ished earlier than its due-date, it would result in storage costs; on the other hand, if a job is finished later than its due-date, itwould violate contractual obligation with the customer. Cheng [33] initiated research on scheduling with due windows. Inmodern manufacturing management, scheduling problems with due window assignment are important issues. Many papershave been conducted on these issues in different scheduling environments. Yin et al. [34] studied a single-machine sched-uling problem with batch delivery cost and an assignable common due window simultaneously. They provided polynomial-time solutions for considered problems. Yeung et al. [35] studied a non-preemptive two-stage flowshop scheduling problemunder the environment of a common due window. The objective is to minimize the earliness and tardiness. They showedthat the problem is NP-complete in the strong sense, and developed a branch and bound algorithm and a heuristic to solvethe problem. For more recent works on scheduling with a common due window, we refer the reader to the comprehensivesurveys by Cheng and Gupta [36], Gordon et al. [37], and the recent papers by Yeung et al. [38], Cheng [39], Baker and Scud-der [40], Liman et al. [41], Yeung et al. [42], Yeung [43], Mosheiov and Sarig [44], Yeung[45], Yang et al. [46], Yin et al. [47]and Yin et al. [48]. In this paper, we assume that every job has its own interval constrained actual processing time, which isrequired to restrict in given interval otherwise earliness or tardiness penalty should be paid. For example, in porcelain man-ufacturing processes, the actual processing time can not exceed a given interval otherwise the porcelain may have qualityflaws.

To the best of our knowledge, however, scheduling with simultaneous consideration of interval constrained actual pro-cessing time and machine maintenance has not been explored. Motivated by these points, in this paper we consider the sin-gle machine scheduling with interval constrained actual processing time and piece-rate maintenance, where the piece-ratemaintenance is proposed by Yu et al. [49]. We explore to find the optimal polynomial algorithm to minimize the total cost,which is assumed to conclude production fee and total earliness and tardiness cost. Moreover, we investigate two specialcases where the actual processing time of each job is assumed to be a product function, and explore to find more effectivesolving algorithm to minimize the total cost for the cases.

This paper is organized as follows. We introduce the notation and terminology of this paper in the next section. In Sec-tion 3, we propose the main results of this paper. In Section 4, we conclude this paper, and suggest some topics for future.

2. Notations and problem formulation

A set J ¼ ½J1; J2; . . . ; Jn� of n independent jobs are partitioned into mþ 1 groups and are all available for processing at timezero. The machine can handle one job at a time. In the manufacturing process, the jobs are non-preemptive. Each job j 2 J isassociated with the normal processing time pj. We extend the general processing time function of [23] and construct a moregeneral model of the actual job processing time

prj ¼ fjðpj; rÞ; ð1 6 j 6 n; 1 6 r 6 njÞ: ð1Þ

Observing from Eq. (1), different jobs may have different actual processing time functions which are not restricted to be spe-cific and monotone.

During the maintenance, the machine is stopped. After the maintenance activity, the machine will revert to its initial con-dition. We assume that the machine need to be maintained one time when every k jobs are completed and maintenance isjust finished at time zero. The jobs will be processed from a group continuously. Thus, we denote the schedule asr=[G1;M1; . . . ;Gi;Mi; . . . ;Mm;Gmþ1], where Gi denotes the ith group and Mi denotes the ith maintenance. Let ni denote thenumber of jobs in the ith group, and m be the maintenance frequency, which is equal to the integer part of n

k, i.e., m ¼ bnkc.We can obtain that

Pmþ1i¼1 ni ¼ n. Let C½l;r� be the completion time of the job scheduled in rth position of the lth group.

Let Gti denote the time length of ith group, and ti be the duration of the maintenance, i.e., ti ¼ aGt

i þ b, where a and b aretwo constants, a P 0. As in Zhao and Tang [13], we denote the group Gi as Gi=[J½i;1�; J½i;2�,. . .,J½i;ni�1�; J½i;ni ��, where J½i;r� is the rth jobin the group Gi.

Let p½l;r� denote the normal processing time of job J½l;r� and pr½l;r� be actual processing time of job J½l;r�, where pr

½l;r�=f½l;r�(p½l;r�,r).Let aj and bj denote the lower limit and upper limit of the interval constraint of every job, respectively. Then, the earliness ofjob j is denoted as Ej, i.e., Ej=maxf0; aj � pr

j g. The tardiness of job j is denoted as Tj, i.e., Tj=max{0; prj � bjg. Let Cmax denote the

makespan, i.e., Cmax=max{Cjjj ¼ 1;2; . . . ;n}, where Cj denotes the completion time of the job j. The total earliness of all jobs isdenoted by

Pnj¼1Ej, and the total tardiness of all jobs is denoted by

Pnj¼1Tj.

In the manufacturing process, the production fee is determined by the length of the working time. Moreover, the earlinesscost and tardiness cost are assumed to be linear relationship with the total earliness and tardiness of all jobs, respectively.Thus, We define the total cost as follows:

TC ¼ aCmax þ bXn

j¼1

Ej þ cXn

j¼1

Tj;

P. Xue et al. / Applied Mathematics and Computation 226 (2014) 415–422 417

where a; b and c are the unit production fee, the unit earliness cost and the unit tardiness cost, respectively. a; b and c shouldbe positive numbers, i.e., a; b; c > 0.

3. The main results

The completion time of all jobs can be obtained as follows:

C ½mþ1;nmþ1 � ¼ ðaþ 1ÞRml¼1R

kr¼1f½l;r�ðp½l;r�; rÞ þ Rnmþ1

r¼1 f½mþ1;r�ðp½mþ1;r�; rÞ þmb: ð2Þ

From the Eq. (2), it can be obtained that

Cmax ¼ C½mþ1;nmþ1 � ¼ ðaþ 1ÞRml¼1R

kr¼1f½l;r�ðp½l;r�; rÞ þ Rnmþ1

r¼1 f½mþ1;r�ðp½mþ1;r�; rÞ þmb: ð3Þ

The total earliness is given by

Rmþ1l¼1 Rni

r¼1E½l;r� ¼ Rml¼1R

kr¼1maxf0; a½l;r� � f½l;r�ðp½l;r�; rÞg þ Rnmþ1

r¼1 maxf0; a½mþ1;r� � f½mþ1;r�ðp½mþ1;r�; rÞg: ð4Þ

Similarly, the total tardiness is given by

Rmþ1l¼1 Rni

r¼1T ½l;r� ¼ Rml¼1R

kr¼1maxf0; f½l;r�ðp½l;r�; rÞ � b½l;r�g þ Rnmþ1

r¼1 maxf0; f½mþ1;r�ðp½mþ1;r�; rÞ � b½mþ1;r�g: ð5Þ

Then, the total cost can be obtained as follows:

TC¼aCmaxþbRnj¼1EjþcRn

j¼1Tj¼aðRml¼1R

kr¼1ðaþ1Þf½l;r�ðp½l;r�;rÞþmbþRnmþ1

r¼1 f½mþ1;r�ðp½mþ1;r�;rÞÞ

þbðRml¼1R

kr¼1maxf0;a½l;r� � f½l;r�ðp½l;r�;rÞgþRnmþ1

r¼1 maxf0;a½mþ1;r� � f½mþ1;r�ðp½mþ1;r�;rÞgÞþcðRm

l¼1Rkr¼1maxf0;f½l;r�ðp½l;r�;rÞ�b½l;r�gþRnmþ1

r¼1 maxf0;f½mþ1;r�ðp½mþ1;r�;rÞ�b½mþ1;r�gÞ¼Rm

l¼1Rkr¼1ðaðaþ1Þf½l;r�ðp½l;r�;rÞþbmaxf0;a½l;r� � f½l;r�ðp½l;r�;rÞgþcmaxf0;f½l;r�ðp½l;r�;rÞ�b½l;r�gÞ

þRnmþ1r¼1 ðaf½mþ1;r�ðp½mþ1;r�;rÞþbmaxf0;a½mþ1;r� � f½mþ1;r�ðp½mþ1;r�;rÞgþcmaxf0;f½mþ1;r�ðp½mþ1;r�;rÞ�b½mþ1;r�gÞþamb: ð6Þ

For simplicity, let b maxf0; a½l;r� � f½l;r�ðp½l;r�; rÞg þ c maxf0; f½l;r�ðp½l;r�; rÞ � b½l;r�g be denoted by Z. Then, we introduce a newfunction f ð1Þ½l;r� ðp½l;r�; rÞ as follows:

f ð1Þ½l;r� ðp½l;r�; rÞ ¼aðaþ 1Þf½l;r�ðp½l;r�; rÞ þ Z; l ¼ 1;2; . . . ;m;

af½l;r�ðp½l;r�; rÞ þ Z; l ¼ mþ 1:

(ð7Þ

Combining (6) and (7), we can obtain

TC½mþ1;nmþ1 � ¼Xmþ1

l¼1

Xni

r¼1

f ð1Þ½l;r� ðp½l;r�; rÞ þ amb:

The problem we studied in this section can be denoted by 1jprj ¼ fjðpj; rÞ; PRM ¼ k; ti ¼ aGt

i þ bjTC. For a given the numberof jobs k, we can get the number of groups by equation m ¼ bnkc. Then, we can obtain that mb is a constant. We explore to finda polynomial to solve the problem. The problem 1jpr

j ¼ fjðpj; rÞ; PRM ¼ k; ti ¼ aGti þ bjTC can be transformed as the following

standard assignment problem:

minXn

j¼1

Xmþ1

l¼1

Xni

r¼1

wjlrxjlr þ amb;

st:Xn

j¼1

xjlr ¼ 1; l ¼ 1;2; . . . ;mþ 1; r ¼ 1;2; . . . ;ni;

Xmþ1

l¼1

Xni

r¼1

xjlr ¼ 1; j ¼ 1;2; . . . ; n;

xjlr ¼ 0 or 1; j ¼ 1;2; . . . ;n; l ¼ 1;2; . . . ;mþ 1; r ¼ 1;2; . . . ni;

ð8Þ

where wjlr ¼ f ð1Þ½l;r� ðp½l;r�; rÞ. We define xjlr ¼ 0 or 1 such that xjlr ¼ 1 if job Jj is scheduled in the rth position in the group Gl andxjlr ¼ 0 otherwise. In case of k ¼ n, all jobs are completed in one group, and the machine does not require maintenance. Thus,

the objective of the assignment problem is notPn

j¼1

Pmþ1l¼1

Pnir¼1wjlrxjlr þ amb, but

Pnj¼1

Pmþ1l¼1

Pnir¼1wjlrxjlr .

In order to minimize the total cost, we propose a polynomial time algorithm to determine jointly the optimal k, and theoptimal job sequence.

Algorithm 1.Step 1. For each k (k=1,2,. . .,n-1,n), solve the assignment problem (8) and calculate the objective value TCðkÞ.Step 2. Let ðTCðkÞÞ�=min ðTCðkÞ, (k=1,2,. . .,n)).

Theorem 1. The 1jprj ¼ fjðpj; rÞ; PRM ¼ k; ti ¼ aGt

i þ bjTC problem can be optimal solved by Algorithm 1 in Oðn4Þ time.

418 P. Xue et al. / Applied Mathematics and Computation 226 (2014) 415–422

Proof. For a fixed the number of jobs k, the problem 1jprj ¼ fjðpj; rÞ; PRM ¼ k; ti ¼ aGt

i þ bjTC can be optimally solved via theassignment problem (8), and the computational complexity of obtaining the optimal job sequence is Oðn3Þ. Since k has n pos-sible values, the computational complexity of solving the problem 1jpr

j ¼ fjðpj; rÞ; PRM ¼ k; ti ¼ aGti þ bjTC is Oðn4Þ. h

In what follows, we investigate two special cases of the 1jprj ¼ fjðpj; rÞ; PRM ¼ k; ti ¼ aGt

i þ bjTC problem, and explore tofind a more efficient algorithm for each case. We assume that pr

½l;r� ¼ p½l;r�ra0 ði:e:; pr

j ¼ pjra0 Þ; ti ¼ bðb > 0Þ and a P b.

The first case concerns the position-dependent learning effect, a0 < 0. In this case, we assume that every job has its ownconstrained interval and has no tardiness. Let pjb0 ð0 < b0 < 1Þ and pj denote the lower limit and upper limit of the con-strained interval of the job j, respectively.

The second case concerns the position-dependent aging effect, a0 > 0. In this case, we assume that every job has its ownconstrained interval and has no earliness. Let pj and pjb0 ðb0 > 1Þ denote the lower limit and upper limit of the constrainedinterval of the job j, respectively.

We first consider the case of the position-dependent learning effect. It is obviously that machine does not need mainte-nance in the optimal scheduling. Then, we can denote the scheduling problem with learning effect as 1jpr

j ¼ pjra0 jTC.

Theorem 2. If a0 < 0, the problem 1jprj ¼ pjr

a0 jTC can be optimally solved by the shortest processing time first (SPT) rule inOðn log nÞ time.

Proof. Assume that an optimal schedule r1 is not the SPT sequence. Then, there must exist a pair of adjacent jobs i and j sothat pi P pj, with job j following job i. We assume that job i is scheduled in the rth position.

Now let a new job schedule r2 be formed from r1 by interchanging jobs i and j and keeping the other jobs in the sameposition as in r1. The exchange of the jobs i and j is illustrated by Fig. 1, where A denotes the set of jobs preceding jobs i and jin both schedules, and B is the set of jobs following jobs i and j in both schedules. Let TCðr1Þ and TCðr2Þ denote the total costof r1 and r2, respectively. Since the positions of the jobs in A and B remain unchanged, the cost of processing and theearliness of the jobs in A and B remains unchanged.

Since not all jobs in the sequence have earliness in case of learning effect, we aim to find the tipping position r�, whoseillustration is proposed in Fig. 2. Let pib0 ¼ pir

a0 , we can obtain r� ¼ bffiffiffiffiffib0

a0pc. In case of r 6 r�, since pib0 6 pir

a0 , it can be seenthat the job before the position (r� þ 1) has no earliness. While in case of r > r�, since pib0 > pir

a0 , it can be seen that the jobbehind the position r� has earliness. In order to prove Theorem 2, we consider the following three cases:

(1) r < r�;(2) r ¼ r�;(3) r > r�.

Case (1). r < r�

In sequence r1 and r2, the jobs i and j have no earliness. Then, it can be obtained

TCðr1Þ � TCðr2Þ ¼ aðpira0 þ pjðr þ 1Þa0 Þ � aðpjr

a0 þ piðr þ 1Þa0Þ ¼ aðpi � pjÞðra0 � ðr þ 1Þa0ÞP 0:

Then, the schedule r1 is dominated by the schedule r2 in this case.Case (2). r ¼ r�

In sequence r1, the job i has no earliness and the job j has earliness, but in the sequence r2 the case is converse. Then, itcan be obtained

TCðr1Þ � TCðr2Þ ¼ ðaðpira0 þ pjðr þ 1Þa0 Þ þ bðpjb0 � pjðr þ 1Þa0 ÞÞ � ðaðpjr

a0 þ piðr þ 1Þa0 Þ þ bðpib0 � piðr þ 1Þa0ÞÞ¼ ðpi � pjÞðaðra0 � ðr þ 1Þa0 Þ � bðb0 � ðr þ 1Þa0ÞÞP 0:

Then, the schedule r1 is dominated by the schedule r2 in this case.

Fig. 1. The illustration of the exchange of jobs Ji and Jj.

Fig. 2. The illustration of the tipping position r� .

P. Xue et al. / Applied Mathematics and Computation 226 (2014) 415–422 419

Case (3). r > r�

In sequence r1 and r2, the jobs i and j have earliness. Then, it can be obtained

TCðr1Þ � TCðr2Þ ¼ ðaðpira0 þ pjðr þ 1Þa0 Þ þ bðpib0 � pir

a0Þ þ bðpjb0 � pjðr þ 1Þa0 ÞÞ � ðaðpjra0 þ piðr þ 1Þa0 Þ

þ bðpjb0 � pjra0Þ þ bðpib0 � piðr þ 1Þa0 ÞÞ ¼ ðra0 � ðr þ 1Þa0Þðpi � pjÞða� bÞP 0:

Then, the schedule r1 is dominated by the schedule r2 in this case.Summing the above argument, in any case, we obtain that the interchange of jobs i and j will result in a decrease in total

cost. By the repeated application of this argument, any sequence that is not a SPT sequence can be improved with respect tothe total cost. Then, the problem 1jpr

j ¼ pjra0 jTC can be optimally solved by the SPT rule in Oðn log nÞ time. h

In what follows, we discuss the case of the position-dependent aging effect. It is necessary to perform maintenance for themachine in the optimal schedule of the 1jpr

j ¼ pjra0 ; PRM ¼ k; ti ¼ bjTC problem. We present a modified Longest Processing

Time first (LPT) rule for the 1jprj ¼ pjra0 ; PRM ¼ k; ti ¼ bjTC problem.

The modified LPT rule. At first, we sequence the jobs in non-increasing order of their normal processing times. Then, sche-dule the job in the first position of each group one by one. If the first position of each group is filled, then schedule theremaining job in the second position of each group one by one. If all the second positions are filled, fill the third position,and so on, until all jobs are scheduled.

Theorem 3. If a0 > 0, the problem 1jprj ¼ pjr

a0 ; PRM ¼ k; ti ¼ bjTC can be optimally solved by the modified LPT rule in Oðn2lognÞtime.

Proof. Assume that an optimal schedule r3 is not sequenced by the modified LPT rule. We consider jobs J½i;r� and J½j;rþ1�. Job J½i;r�is sequenced in rth position of the group Gi. Job J½j;rþ1� is sequenced in ðr þ 1Þth position of the group Gj. p½i;r� and p½j;rþ1� denotethe normal processing times of jobs J½i;r� and J½j;rþ1�, respectively. Moreover, assume that p½i;r� 6 p½j;rþ1�.

Now let a new job schedule r4 be formed from r3 by interchanging jobs J½i;r� and J½j;rþ1� and keeping the other jobs in thesame position as in r3. The exchange of the jobs J½i;r� and J½j;rþ1� is illustrated by Fig. 3, where p1;p2 and p3 are partial scheduleof the schedule r3. Let TCðr3Þ and TCðr4Þ denote the total cost of r3 and r4, respectively. For the reason that the positions ofthe jobs in p1;p2 and p3 unchanged, the cost of processing and the tardiness of every job in p1;p2 and p3 remainsunchanged.

Since not all jobs in the sequence have tardiness in case of aging effect, so we aim to find the tipping position r�, whose

illustration is proposed in Fig. 4. Let pib0 ¼ pira0 , we can obtain r� ¼ b

ffiffiffiffiffib0

a0pc. In case of r 6 r�, since pib0 P pir

a0 , the job beforethe position (r� þ 1) has no tardiness. While in case of r > r�, since pib0 < pir

a0 , the job behind the position r� has tardiness. Inorder to prove Theorem 3, we consider the following three cases:

(1) r < r�;(2) r ¼ r�;(3) r > r�.

Fig. 3. The illustration of the exchange of jobs J½i;r� and J½j;rþ1� .

Fig. 4. The illustration of the tipping position r� .

420 P. Xue et al. / Applied Mathematics and Computation 226 (2014) 415–422

case (1). r < r�

In sequence r3 and r4, the jobs J½i;r� and J½j;rþ1� have no tardiness. Then, it can be obtained

TCðr3Þ � TCðr4Þ ¼ aðp½i;r�ra0 þ p½j;rþ1�ðr þ 1Þa0 Þ � aðp½j;rþ1�ra0 þ p½i;r�ðr þ 1Þa0 Þ ¼ aðp½i;r� � p½j;rþ1�Þðra0 � ðr þ 1Þa0 ÞP 0:

Then, the schedule r3 is dominated by the schedule r4.case (2). r ¼ r�

In sequence r3, the job J½i;r� has no tardiness and the job J½j;rþ1� has tardiness, but in the sequence r4 the case is converse.Then, it can be obtained

TCðr3Þ � TCðr4Þ ¼ ðaðp½i;r�ra0 þ p½j;rþ1�ðr þ 1Þa0 Þ þ cðp½j;rþ1�ðr þ 1Þa0 � p½j;rþ1�b0ÞÞ � ðaðp½j;rþ1�ra0 þ p½i;r�ðr þ 1Þa0 Þ

þ cðp½i;r�ðr þ 1Þa0 � p½i;r�b0ÞÞ¼ ðp½i;r� � p½j;rþ1�Þðaðra0 � ðr þ 1Þa0 Þ þ rðb0 � ðr þ 1Þa0ÞÞP 0:

Then, the schedule r4 dominates the schedule r3 in this case.case (3). r > r�

In sequence r3 and r4, the jobs J½i;r� and J½j;rþ1� have tardiness. Then, it can be obtained

TCðr3Þ � TCðr4Þ ¼ ðaðp½i;r�ra0 þ p½j;rþ1�ðr þ 1Þa0 Þ þ cðp½i;r�ra0 � p½i;r�b0Þ þ cðp½j;rþ1�ðr þ 1Þa0 � p½j;rþ1�b0ÞÞ � ðaðp½j;rþ1�ra0

þ p½i;r�ðr þ 1Þa0 Þ þ cðp½j;rþ1�ra0 � p½j;rþ1�b0Þ þ cðp½i;r�ðr þ 1Þa0 � p½i;r�b0ÞÞ

¼ ðra0 � ðr þ 1Þa0Þðp½i;r� � p½j;rþ1�Þðaþ cÞP 0:

Then, the schedule r3 is dominated by the schedule r4 in this case.Based on the above argument, we obtain that the interchange of jobs J½i;r� and J½j;rþ1� will result in a decrease in total cost.

By the repeated application of this argument, any sequence that is not a modified LPT sequence can be improved with respectto the total cost. For a fixed k, the 1jpr

j ¼ pjra0 ; PRM ¼ k; ti ¼ bjTC problem can be optimally solved by the modified LPT rule,

and the computational complexity is Oðn log nÞ. Since kðk ¼ 1;2; . . . ;nÞ has n possible values, the total computationalcomplexity of solving the problem 1jpr

j ¼ pjra0 ; PRM ¼ k; ti ¼ bjTC is Oðn2lognÞ. h

4. Conclusions

The paper considered the scheduling problem with piece-rate machine maintenance and interval constrained actual pro-cessing time. The job actual processing time is required to restrict in given interval otherwise earliness or tardiness penaltyshould be paid. The objective is to minimize the total cost that is a linear function of the makespan, the earliness and tar-diness penalties of all jobs. For scheduling problem with general position-dependent processing times, we showed thatthe total cost minimization problem can be optimally solved in Oðn4Þ time. Moreover, for the two special cases, we showedthat the total cost minimization problem with learning effect can be solved by SPT rule in Oðn log nÞ time, and the total costminimization problem with aging effect can be solved by the modified LPT rule in Oðn2lognÞ time. For the future research, it issuggested to investigate the scheduling problem with piece-rate machine maintenance and interval constrained actual pro-cessing time in the context of parallel machines scheduling problems or job-shop scheduling problems.

Acknowledgements

The authors wish to thank editors and anonymous referees for their helpful comments. This work is supported byNational Nature Science Foundation Project of China (71171046) and the Scientific Research Foundation of Graduate Schoolof Southeast University (YBJJ1239).

References

[1] A. Bachman, T.C.E. Cheng, J. Janiak, C.T. Ng, Scheduling start time dependent jobs to minimize the total weighted completion time, Journal of theOperational Research Society 53 (2002) 688–693.

P. Xue et al. / Applied Mathematics and Computation 226 (2014) 415–422 421

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

[3] M. Ji, T.C.E. Cheng, Scheduling resumable simple linear deteriorating jobs on a single machine with an availability constraint to minimize makespan,Computers and Industrial Engineering 59 (2010) 794–798.

[4] G. Mosheiov, O. Daniel, Minimizing the number of tardy jobs on a proportionate flowshop with general position-dependent processing times,Computers & Operations Research 39 (2012) 1601–1604.

[5] C.J. Hsu, S.J. Yang, D.L. Yang, Two due date assignment problems with position-dependent processing time on a single-machine, Computers andIndustrial Engineering 60 (2011) 796–800.

[6] T.C.E. Cheng, G. Wang, Single machine scheduling with learning effect considerations, Annals of Operations Research 98 (2000) 273–290.[7] A. Bachman, A. Janiak, Scheduling jobs with position-dependent processing times, Journal of the Operational Research Society 55 (2004) 257–264.[8] J.B. Wang, Z.Q. Xia, Flow shop scheduling with deteriorating jobs under dominating machines, OMEGA – The International Journal of Management

Science 34 (2006) 327–336.[9] D. Biskup, Single-machine scheduling with learning considerations, European Journal of Operational Research 115 (1999) 173–178.

[10] W. Kuo, D.L. Yang, Minimizing the makespan in a single machine scheduling problem with the cyclic process of an aging effect, Journal of theOperational Research Society 59 (2008) 416–420.

[11] S.J. Yang, D.L. Yang, T.C.E. Cheng, Single-machine due-window assignment and scheduling with job-dependent aging effects and deterioratingmaintenance, Computers & Operations Research 37 (2010) 1510–1514.

[12] S.J. Yang, D.L. Yang, Minimizing the makespan on single-machine scheduling with aging effect and variable maintenance activities, OMEGA – TheInternational Journal of Management Science 38 (2010) 528–533.

[13] C.L. Zhao, H.Y. Tang, Single machine scheduling with general job-dependent aging effect and maintenance activities to minimize makespan, AppliedMathematical Modelling 34 (2010) 837–841.

[14] W.C. Lee, A note on deteriorating jobs and learning in single-machine scheduling problems, International Journal of Business and Economics 3 (2004)83–89.

[15] T.C.E. Cheng, C.C. Wu, W.C. Lee, Some scheduling problems with deteriorating jobs and learning effects, Computers and Industrial Engineering 54(2008) 972–982.

[16] J.B. Wang, Single-machine scheduling problems with the effects of learning and deterioration, OMEGA – The International Journal of ManagementScience 35 (2007) 397–402.

[17] J.B. Wang, T.C.E. Cheng, Scheduling problems with the effects of deterioration and learning, Asia-Pacific Journal of Operational Research 24 (2007) 1–17.

[18] T.C.E. Cheng, W.C. Lee, Scheduling problems with deteriorating jobs and learning effects including proportional setup times, Computers and IndustrialEngineering 58 (2010) 326–331.

[19] Y.Q. Yin, D.H. Xu, K.B. Sun, H.X. Li, Some scheduling problems with general position-dependent and time-dependent learning effects, InformationScience 179 (2009) 2416–2425.

[20] P.J. Lai, W.C. Lee, Single-machine scheduling with general sum-of-processing-time-based and position-based learning effects, OMEGA – TheInternational Journal of Management Science 39 (2011) 467–471.

[21] G. Mosheio, J. Sidney, Scheduling with general job-dependent learning curves, European Journal of Operational Research 147 (2003) 665–670.[22] J.B. Wang, M.Z. Wang, Worst-case behavior of simple sequencing rules in flow shop scheduling with general position-dependent learning effects,

Annals of Operations Research 191 (2011) 155–169.[23] G. Mosheiov, Proportionate flowshops with general position-dependent processing times, Information Processing Letters 111 (2011) 174–177.[24] M.A. Kubzin, V.A. Strusevich, Planning machine maintenance in two-machine shop scheduling, Operations Research 54 (2006) 789–800.[25] G. Mosheiov, J. Sidney, Scheduling a deteriorating maintenance activity on a single machine, Journal of the Operational Research Society 61 (2010)

867–882.[26] T.C.E. Cheng, C.J. Hsu, D.L. Yang, Unrelated parallel-machine scheduling with deteriorating maintenance activities, Computers and Industrial

Engineering 60 (2011) 602–605.[27] B. Mor, G. Mosheiov, Scheduling a maintenance activity and due-window assignment based on common flow-allowance, International Journal of

Production Economics 135 (2012) 222–230.[28] T.C.E. Cheng, S.J. Yang, D.L. Yang, Common due-window assignment and scheduling of linear time-dependent deteriorating jobs and a deteriorating

maintenance activity, International Journal of Production Economics 135 (2012) 154–161.[29] C.J. Liao, W.J. Chen, Single machine scheduling with periodic maintenance and nonresumable jobs, Computers & Operations Research 30 (2003) 1347–

1353.[30] M. Ji, Y. He, T.C.E. Cheng, Single machine scheduling with periodic maintenance to minimize makespan, Computers & Operations Research 34 (2007)

1764–1770.[31] M. Ji, T.C.E. Cheng, Scheduling with job-dependent learning effects and multiple rate-modifying activities, Information Processing Letters 110 (2010)

460–463.[32] D.L. Yang, T.C.E. Cheng, S.J. Yang, C.J. Hsu, Unrelated parallel-machine scheduling with aging effects and multi-maintenance activities, Computers &

Operations Research 39 (2012) 1458–1464.[33] T.C.E. Cheng, Optimal common due-date with limited completion time deviation, Computers & Operations Research 15 (1988) 91–96.[34] Y.Q. Yin, T.C.E. Cheng, C.J. Hsu, C.C. Wu, Single-machine batch delivery scheduling with an assignable common due window, OMEGA – The

International Journal of Management Science 41 (2013) 216–225.[35] W.K. Yeung, C. Oguz, T.C.E. Cheng, Two-stage flowshop earliness and tardiness machine scheduling involving a common due window, International

Journal of Production Economics 90 (2004) 421–434.[36] T.C.E. Cheng, M.C. Gupta, Survey of scheduling research involving due-date determination decisions, European Journal of Operational Research 38

(1989) 156–166.[37] V.S. Gordon, J.M. Proth, C. Chu, A survey of the state-of-art of common due date assignment and scheduling research, European Journal of Operational

Research 139 (2002) 1–25.[38] W.K. Yeung, C. Oguz, T.C.E. Cheng, Single machine scheduling with a common due window, Computers & Operations Research 28 (2001) 157–175.[39] T.C.E. Cheng, A heuristic for common due-date assignment and job scheduling on parallel machines, Journal of the Operational Research Society 40

(1989) 1129–1135.[40] K.R. Baker, G.D. Scudder, Sequencing with earliness and tardiness penalties: a review, Operations Research 38 (1990) 22–35.[41] S.D. Liman, S.S. Panwalkar, S. Thongmee, Common due window size and location determination in a single machine scheduling problem, Journal of the

Operational Research Society 49 (1998) 1007–1010.[42] W.K. Yeung, C. Oguz, T.C.E. Cheng, Minimizing weighed number of early and tardy jobs with a common due window involving location penalty, Annals

of Operations Research 108 (2001) 33–54.[43] W.K. Yeung, C. Oguz, T.C.E. Cheng, Two-machine flow shop scheduling with due window to minimize weighted number of early and tardy jobs, Naval

Research Logistics 56 (2009) 593–599.[44] G. Mosheiov, A. Sarig, Scheduling a maintenance activity and due-window assignment on a single machine, Computers & Operations Research 36

(2009) 2541–2545.[45] W.K. Yeung, T.M. Choi, T.C.E. Cheng, Optimal scheduling of a single-supplier single-manufacturer supply chain with common due windows, IEEE

Transactions on Automatic Control 55 (2010) 2767–2777.

422 P. Xue et al. / Applied Mathematics and Computation 226 (2014) 415–422

[46] S.J. Yang, D.L. Yang, T.C.E. Cheng, Single-machine due-window assignment and scheduling with job-dependent aging effects and deterioratingmaintenance, Computers & Operations Research 37 (2010) 1510–1514.

[47] Y.Q. Yin, T.C.E. Cheng, J.Y. Wang, C.C. Wu, Single-machine common due window assignment and scheduling to minimize the total cost, DiscreteOptimization 10 (2013) 42–53.

[48] Y.Q. Yin, T.C.E. Cheng, S.R. Cheng, C.C. Wu, Single-machine due window assignment and scheduling with a common flow allowance and controllablejob processing times, Journal of the Operational Research Society 64 (2013), http://dx.doi.org/10.1057/jors.2012.161.

[49] X.Y. Yu, Y.L. Zhang, D.H. Xu, Y.Q. Yin, Single machine scheduling problem with two synergetic agents and piece-rate maintenance, AppliedMathematical Modelling 37 (2013) 1390–1399.