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Applied Mathematical Modelling 38 (2014) 384–391
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Applied Mathematical Modelling
journal homepage: www.elsevier .com/locate /apm
Short communication
Single-machine ready times scheduling with group technologyand proportional linear deterioration
0307-904X/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.apm.2013.05.064
⇑ Corresponding author at: School of Science, Shenyang Aerospace University, Shenyang 110136, China. Tel.: +86 24 89723661.E-mail address: [email protected] (X. Huang).
Yang-Tao Xu a,b, Yu Zhang c, Xue Huang b,d,⇑a School of Economics and Management, Shenyang Aerospace University, Shenyang 110136, Chinab State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710053, Chinac School of Foreign Language, Shenyang Aerospace University, Shenyang 110136, Chinad School of Science, Shenyang Aerospace University, Shenyang 110136, China
a r t i c l e i n f o
Article history:Received 19 September 2011Received in revised form 7 May 2013Accepted 31 May 2013Available online 25 June 2013
Keywords:SchedulingSingle machineGroup technologyDeteriorating jobsReady timeHeuristic algorithm
a b s t r a c t
Scheduling research has increasingly taken the concept of deterioration into consideration.In this paper, we study a single machine group scheduling problem with deteriorationeffect, where the jobs are already put into groups, before any optimization. We assume thatthe actual processing times of jobs are increasing functions of their starting times, i.e., thejob processing times are described by a function which is proportional to a linear functionof time. The setup times of groups are assumed to be fixed and known. For some specialcases of minimizing the makespan with ready times of the jobs, we show that the problemcan be solved in polynomial time for the proposed model. For the general case, a heuristicalgorithm is proposed, and the computational experiments show that the performance ofthe heuristic is fairly accurately in obtaining near-optimal solutions. The results imply thatthe average percentage error of the proposed heuristic algorithm from optimal solutions isless than 3%.
� 2013 Elsevier Inc. All rights reserved.
1. Introduction
In recent years, a growing interest in studying scheduling problems with time-dependent processing times (deterioratingjobs), have been award of, i.e., jobs whose processing times are increasing functions of their starting times. Job deteriorationappears, e.g., in scheduling maintenance jobs, steel production, cleaning assignments or emergency medicine, where the jobsthat are processed later will take longer times to process. An extensive survey of different scheduling problems with time-dependent processing times (deteriorating jobs) can be found in Gawiejnowicz [1]. For details on time-dependent schedulingproblems without group technology assumption, the reader may refer to the papers by Wang and Guo [2], Wang and Wang[3], Zhao and Tang [4], Moslehi and Jafari [5], Wang et al. [6], Sun et al. [7], Wang et al. [8], Wei et al. [9], Wang and Wang[10], Wang et al. [11], Wang and Wang [12], Rachaniotis and Pappis [13], Voutsinas and Pappis [14], Huang and Wang [15],Ng et al. [16], Sun et al. [17], Wang et al. [18], and Wang and Wang [19].
In the second place, in group technology, it is conventional to schedule continuously all jobs from the same group. Grouptechnology that groups similar products into families helps increase the efficiency of operations and decrease the require-ment of facilities (Mitrofanov [20], Potts and Van Wassenhove [21], Kovalyov and Janiak [22], Webster and Baker [23]).To the best of our knowledge, only few results concerning scheduling models and problems with time-dependent processingtimes (deteriorating jobs) and group technology simultaneously are known. Wang et al. [24] considered single machine
Y.-T. Xu et al. / Applied Mathematical Modelling 38 (2014) 384–391 385
group scheduling where the setup times of groups are constant, and the actual processing time of a job is a general lineardecreasing function of its starting time. They showed that the makespan minimization problem and total completion timeminimization problem can be solved in polynomial time. Wu et al. [25], Wu and Lee [26], Wang et al. [27], and Wang et al.[28] considered a situation where the group setup times and job processing times are both described by a linear deteriora-tion function. For the simple linear deterioration, Wu et al. [25] proved that the makespan minimization problem and thetotal completion minimization problem can be solved in polynomial time. For the linear deterioration function with identicaldeterioration rate, Wu and Lee [26] proved that the makespan minimization problem remains polynomially solvable. Theyalso showed that the sum of completion times problem is polynomially solvable when the numbers of jobs in each group areequal. For the proportional linear deterioration, Wang et al. [27] proved that the makespan minimization problem and thetotal weighted completion time minimization problem can be solved in polynomial time. For a general deterioration func-tion, Wang et al. [28] proved that the makespan minimization problem can be solved in polynomial time. Huang et al. [29]and Bai et al. [30] considered single machine scheduling problems with learning effects and deteriorating jobs. Wang et al.[31] considered a single machine group scheduling problem, in which the processing time of a job is a simple linear functionof its starting time, and the setup time of a group is assumed to be known and fixed. For a special case, they proved that themakespan minimization problem with ready times of the jobs is solvable in polynomial time.
In this paper, we continue the work of Wang et al. [31], by considering a more general deterioration model that includesthe one given in [31] as a special case. The rest of the paper is organized as follows: In Section 2 we formulate the model. InSections 3 we consider some special cases of the makespan minimization problem and solve the problem in polynomial time.In Section 4, we propose a heuristic algorithm for the general case and give the computational experiments. In the final sec-tion, we conclude the paper.
2. The model
We will use the following notation:
mðm P 2Þ
the number of groups n the total number of jobs Gi group i; i ¼ 1;2; . . . ;m ni the number of jobs belonging to groupGi
Jij
job j in groupGi; i ¼ 1;2; . . . ;m; j ¼ 1;2; . . . ;nirij
the ready (arrival) time of jobJij; i ¼ 1;2; . . . ;m; j ¼ 1;2; . . . ;nisi
the setup time of group Gipij
the actual processing time of job Jijaij
the deterioration rate of job Jijp
a job sequence of n jobs Cij ¼ CijðpÞ the completion time of job Jij in p Cmax ¼maxfCijji ¼ 1;2; . . . ;m; j ¼ 1;2; . . . ;nig the makespan of a given permutationThere are n jobs grouped into m groups, and our problem is to schedule these n jobs on a single machine in order to min-imize the performance measure of makespan (i.e., the maximum completion time of all jobs). We assume that the processingof a job may not be interrupted. Let ni be the number of jobs belonging to group Gi (n1 þ n2 þ . . .þ nm ¼ n). In addition, Jij
denotes the jth job in group Gi, rij P 0 denotes the ready (arrival) time of job Jij; pij denotes the actual processing time ofjob Jij; si denotes the setup time of group Gi required if the machine switches from one group to another and all setup timesof groups for processing at time t0 P 0. Note that the groups’ set up times are independent of the previous and the nextgroup, if any. As in Kononov and Gawiejnowicz [32], we consider the following proportional linear deterioration model
pij ¼ aijðaþ btÞ;
where a P 0; b P 0;aij is the deterioration rate of job Jij, and t is its start time.Let GT indicate that the problem is a scheduling problem with group technology. Adopting the three-field notation for
scheduling problem introduced by Graham et al. [33], we denote the above problem as 1jrij; pij ¼ aijðaþ btÞ; si;GTjCmax.
3. Makespan minimization scheduling problem for some special cases
In this section, we consider a single machine makespan minimization scheduling problem under the proposed model. Forsome special cases, we will show that the problem can be solved in polynomial time.
386 Y.-T. Xu et al. / Applied Mathematical Modelling 38 (2014) 384–391
Theorem 1. For the 1jrij; pij ¼ aijðaþ btÞ; si;GTjCmax problem, if the schedule of groups is given, then an optimal schedule musthave jobs within each group in order of nondecreasing rij; i ¼ 1;2; . . . ;m; j ¼ 1;2; . . . ;ni.
Proof. Suppose that p ¼ ðS1; Jiu; Jiv ; S2Þ and p0 ¼ ðS1; Jiv ; Jiu; S2Þ are two job schedules, where S1 and S2 denote a partialsequence. Let A denote the completion time of the last job in S1. Under p, we have
CiuðpÞ ¼max A; riuf g þ aiuðaþ b maxfA; riugÞ
¼max Aþ ab
� �1þ baiuð Þ; riu þ
ab
� �ð1þ baiuÞ
n o� a
b
ð1Þ
and
CivðpÞ ¼max Ciu; rivf g þ aivðaþ b maxfCiu; rivgÞ
¼max Aþ ab
� �ð1þ baiuÞð1þ baivÞ; riu þ
ab
� �ð1þ baiuÞð1þ baivÞ; riv þ
ab
� �ð1þ baivÞ
n o� a
b:
ð2Þ
Similarly, under p0, we have
Civðp0Þ ¼max Aþ ab
� �ð1þ baivÞ; riv þ
ab
� �ð1þ baivÞ
n o� a
bð3Þ
and
Ciuðp0Þ ¼max Aþ ab
� �ð1þ baivÞð1þ baiuÞ; riv þ
ab
� �ð1þ baivÞð1þ baiuÞ; riu þ
ab
� �ð1þ baiuÞ
n o� a
b: ð4Þ
Suppose riu 6 riv , based on Eqs. (2) and (4), we have
Ciuðp0Þ � CivðpÞ ¼ max Aþ ab
� �ð1þ baiuÞð1þ baivÞ; riv þ
ab
� �ð1þ baiuÞð1þ baivÞ; riu þ
ab
� �ð1þ aiuÞ
n o�max Aþ a
b
� �ð1þ baiuÞð1þ baivÞ; riu þ
ab
� �ð1þ baiuÞð1þ baivÞ; riv þ
ab
� �ð1þ baivÞ
n oP max Aþ a
b
� �ð1þ aiuÞð1þ baivÞ; riv þ
ab
� �ð1þ baiuÞð1þ baivÞ; riu þ
ab
� �ð1þ baiuÞ
n o�max Aþ a
b
� �ð1þ baiuÞð1þ baivÞ; riv þ
ab
� �ð1þ baiuÞð1þ baivÞ; riv þ
ab
� �ð1þ baivÞ
n o¼ 0:
Hence, we have Ciuðp0ÞP Civ ðpÞ. It implies that job processed after Jiu and Jiv under p0 has a later starting time than the samejob under p. Thus, the makespan of all jobs under p is non-greater than that under p0. h
Next, we assume that B denotes the completion time of the ði� 1Þth group and rið1Þ 6 rið2Þ 6 � � � 6 riðniÞ is satisfied for groupGi. Then, we have
CiðniÞðGiÞ ¼max Bþ si þab
� �Yni
l¼1
ð1þ baiðlÞÞ; rið1Þ þab
� �Yni
l¼1
ð1þ baiðlÞÞ; rið2Þ þab
� �Yni
l¼2
ð1þ baiðlÞÞ; . . . ; riðniÞ þab
� �ð1þ baiðniÞÞ
( )
� ab¼max Bþ si þ
ab;
rihðkÞ þ abQhðkÞ�1
l¼1 ð1þ baiðlÞÞ
( )Yni
l¼1
ð1þ baiðlÞÞ �ab; ð5Þ
where ðrihðkÞ þ abÞQni
l¼hðkÞð1þ baiðlÞÞ ¼ maxfðrið1Þ þ abÞQni
l¼1ð1þ baiðlÞÞ; ðrið2Þ þ abÞQni
l¼2ð1þ baiðlÞÞ; . . . ; ðriðniÞ þ abÞð1þ baiðniÞÞg;
hðkÞ 2 f1;2; . . . ;nig. In addition, we assume that ðrihðkÞ þ abÞQni
l¼hðkÞð1þ baiðlÞÞ; si and siQni
l¼1ð1þ bailÞ are agreeable, i.e., ifðrihðkÞ þ a
bÞQni
l¼hðkÞð1þ baiðlÞÞ 6 ðrjhðkÞ þ abÞQnj
l¼hðkÞð1þ bajðlÞÞ it implies that si 6 sj and siQni
l¼1ð1þ bailÞP sjQnj
l¼1ð1þ bajlÞ.
Theorem 2. For the 1jrij; pij ¼ aijðaþ btÞ; si;GTjCmax problem, if ðrihðkÞ þ abÞQni
l¼hðkÞð1þ baiðlÞÞ; si and siQni
l¼1ð1þ bailÞ are agreeable,i.e., if ðrihðkÞ þ a
bÞQni
l¼hðkÞð1þ baiðlÞÞ 6 ðrjhðkÞ þ abÞQnj
l¼hðkÞð1þ bajðlÞÞ implies that si 6 sj and siQni
l¼1ð1þ bailÞP sjQnj
l¼1ð1þ bajlÞ for allthe groups Gi and Gj, then an optimal schedule must have groups sequenced in nondecreasing order of
rihðkÞ þ abQhðkÞ�1
l¼1 ð1þ baiðlÞÞ;
where ðrihðkÞ þ abÞQni
l¼hðkÞð1þ baiðlÞÞ ¼maxfðrið1Þ þ abÞQni
l¼1ð1þ baiðlÞÞ; ðrið2Þ þ abÞQni
l¼2ð1þ baiðlÞÞ; � � � ; ðriðniÞ þ abÞð1þ baiðniÞÞg; hðkÞ 2
f1;2; � � � ;nig, andQ0
l¼1ð1þ baiðlÞÞ ¼ 1.
Proof. Let p ¼ ½S1;Gi;Gj; S2�;p0 ¼ ½S1;Gj;Gi; S2�, where S1 and S2 are partial sequences. Furthermore, we assume that B denotethe completion time of the last job in S1. To show p dominates p0, it suffices to show that CjðnjÞðpÞ 6 CiðniÞðp0Þ. Under p, usingEq. (5), we have
Y.-T. Xu et al. / Applied Mathematical Modelling 38 (2014) 384–391 387
CiðniÞðpÞ ¼max Bþ si þab;
rihðkÞ þ abQhðkÞ�1
l¼1 ð1þ baiðlÞÞ
( )Yni
l¼1
ð1þ baiðlÞÞ �ab
and
CjðnjÞðpÞ ¼max CiðniÞðpÞ þ sj þab;
rjhðkÞ þ abQhðkÞ�1
l¼1 ð1þ bajðlÞÞ
( )Ynj
l¼1
ð1þ aiðlÞÞ �ab
¼max Bþ si þab
� �Yni
l¼1
ð1þ baiðlÞÞ þ sj;rihðkÞ þ a
bQhðkÞ�1l¼1 ð1þ baiðlÞÞ
Yni
l¼1
ð1þ baiðlÞÞ þ sj;rjhðkÞ þ a
bQhðkÞ�1l¼1 ð1þ bajðlÞÞ
( )Ynj
l¼1
ð1þ ajðlÞÞ �ab:
ð6Þ
Similarly, under p0, we have
CjðnjÞðp0Þ ¼max Bþ sj þ
ab;
rjhðkÞ þ abQhðkÞ�1
l¼1 ð1þ bajðlÞÞ
( )Ynj
l¼1
ð1þ bajðlÞÞ �ab
and
CiðniÞðp0Þ ¼ max Bþ sj þ
ab
� �Ynj
l¼1
ð1þ bajðlÞÞ þ si;rjhðkÞ þ a
bQhðkÞ�1l¼1 ð1þ bajðlÞÞ
Ynj
l¼1
ð1þ bajðlÞÞ þ si;rihðkÞ þ a
bQhðkÞ�1l¼1 ð1þ baiðlÞÞ
( )Yni
l¼1
ð1þ aiðlÞÞ �ab:
ð7Þ
If
rihðkÞ þab
� � Yni
l¼hðkÞð1þ baiðlÞÞ 6 rjhðkÞ þ
ab
� � Ynj
l¼hðkÞð1þ bajðlÞÞ;
it implies that si 6 sj and siQni
l¼1ð1þ bailÞP sjQnj
l¼1ð1þ bajlÞ.Stem from Eqs. (6) and (7), we have
CjðnjÞðpÞ � CiðniÞðp0Þ
¼max Bþ si þab
� �Yni
l¼1
ð1þ baiðlÞÞYnj
l¼1
ð1þ bajðlÞÞ þ sj
Ynj
l¼1
ð1þ bajðlÞÞ;rihðkÞ þ a
bQhðkÞ�1l¼1 ð1þ baiðlÞÞ
Yni
l¼1
ð1þ baiðlÞÞYnj
l¼1
ð1þ bajðlÞÞ(
þ sj
Ynj
l¼1
ð1þ bajðlÞÞ;rjhðkÞ þ a
bQhðkÞ�1l¼1 ð1þ bajðlÞÞ
Ynj
l¼1
ð1þ bajðlÞÞ)
�max Bþ sj þab
� �Ynj
l¼1
ð1þ bajðlÞÞYni
l¼1
ð1þ baiðlÞÞ þ si
Yni
l¼1
ð1þ baiðlÞÞ;rjhðkÞ þ a
bQhðkÞ�1l¼1 ð1þ bajðlÞÞ
Ynj
l¼1
ð1þ bajðlÞÞYni
l¼1
ð1þ baiðlÞÞ(
þ si
Yni
l¼1
ð1þ baiðlÞÞ;rihðkÞ þ a
bQhðkÞ�1l¼1 ð1þ baiðlÞÞ
Yni
l¼1
ð1þ baiðlÞÞ)
6 max Bþ sj þab
� �Yni
l¼1
ð1þ baiðlÞÞYnj
l¼1
ð1þ bajðlÞÞ þ si
Yni
l¼1
ð1þ baiðlÞÞ;rjhðkÞ þ a
bQhðkÞ�1l¼1 ð1þ bajðlÞÞ
Yni
l¼1
ð1þ baiðlÞÞYnj
l¼1
ð1þ bajðlÞÞ(
þ si
Yni
l¼1
ð1þ baiðlÞÞ;rjhðkÞ þ a
bQhðkÞ�1l¼1 ð1þ bajðlÞÞ
Ynj
l¼1
ð1þ bajðlÞÞ)
�max Bþ sj þab
� �Ynj
l¼1
ð1þ bajðlÞÞYni
l¼1
ð1þ baiðlÞÞ þ si
Yni
l¼1
ð1þ baiðlÞÞ;rjhðkÞ þ a
bQhðkÞ�1l¼1 ð1þ bajðlÞÞ
Ynj
l¼1
ð1þ bajðlÞÞYni
l¼1
ð1þ baiðlÞÞ(
þ si
Yni
l¼1
ð1þ baiðlÞÞ;rihðkÞ þ a
bQhðkÞ�1l¼1 ð1þ baiðlÞÞ
Yni
l¼1
ð1þ baiðlÞÞ)¼ 0:
Therefore, the theorem is proved. h
Corollary 1. For the 1jrij; pij ¼ aijðaþ btÞ; si ¼ s > 0;GTjCmax problem, an optimal schedule must have:
1. Jobs within each group sequenced in order of nondecreasing rij, i.e.,
rið1Þ 6 rið2Þ 6 . . . 6 riðniÞ; i ¼ 1;2; . . . ;m;
388 Y.-T. Xu et al. / Applied Mathematical Modelling 38 (2014) 384–391
2. If rihðkÞ þ ab
� �Qnil¼hðkÞð1þ baiðlÞÞ and
Qnil¼1ð1þ bailÞ are agreeable, i.e., if ðrihðkÞ þ a
bÞQni
l¼hðkÞð1þ baiðlÞÞ 6 ðrjhðkÞ þ abÞQnj
l¼hðkÞð1þ bajðlÞÞimplies that
Qnil¼1ð1þ bailÞP
Qnj
l¼1ð1þ bajlÞ for all the groups Gi and Gj, then the groups sequenced in nondecreasing order of
rihðkÞ þ abQhðkÞ�1
l¼1 ð1þ baiðlÞÞ;
where ðrihðkÞ þ abÞQni
l¼hðkÞð1þ baiðlÞÞ ¼maxfðrið1Þ þ abÞQni
l¼1ð1þ baiðlÞÞ; ðrið2Þ þ abÞQni
l¼2ð1þ baiðlÞÞ; . . . ; ðriðniÞ þ abÞð1þ baiðniÞÞg;
hðkÞ 2 f1;2; . . . ;nig.
Corollary 2. For the 1jrij; pij ¼ aijðaþ btÞ; si ¼ 0;GTjCmax problem, an optimal schedule must have:
1. Jobs within each group sequenced in order of nondecreasing rij, i.e.,
rið1Þ 6 rið2Þ 6 � � � 6 riðniÞ; i ¼ 1;2; . . . ;m;
2. The groups are arranged in nondecreasing order of
rihðkÞ þ abQhðkÞ�1
l¼1 ð1þ baiðlÞÞ;
where ðrihðkÞ þ abÞQni
l¼hðkÞð1þ baiðlÞÞ ¼ maxfðrið1Þ þ abÞQni
l¼1ð1þ baiðlÞÞ; ðrið2Þ þ abÞQni
l¼2ð1þ baiðlÞÞ; . . . ; ðriðniÞ þ abÞð1þ baiðniÞÞg;
hðkÞ 2 f1;2; . . . ;nig.
From Theorem 1 and Theorem 2, if ðrihðkÞ þ abÞQni
l¼hðkÞð1þ baiðlÞÞ, si and siQni
l¼1ð1þ bailÞ are agreeable, i.e., ifðrihðkÞ þ a
bÞQni
l¼hðkÞð1þ baiðlÞÞ 6 ðrjhðkÞ þ abÞQnj
l¼hðkÞð1þ bajðlÞÞ implies that si 6 sj and siQni
l¼1ð1þ bailÞP sjQnj
l¼1ð1þ bajlÞ for all thegroups Gi and Gj, then the problem 1jrij; pij ¼ aijðaþ btÞ; si;GTjCmax can be solved by the following algorithm:
Algorithm 1.
Step 1. Arrange the jobs within each group in nondecreasing order of rij, i.e.,
rið1Þ 6 rið2Þ 6 � � � 6 riðniÞ; i ¼ 1;2; . . . ;m:
Step 2. LetðrihðkÞ þ a
bÞQni
l¼hðkÞð1þ baiðlÞÞ ¼maxfðrið1Þ þ abÞQni
l¼1ð1þ baiðlÞÞ; ðrið2Þ þ abÞQni
l¼2ð1þ baiðlÞÞ; . . . ; ðriðniÞ þ abÞð1þ baiðniÞÞg,
hðkÞ 2 f1;2; . . . ;nig, calculate hðkÞ and rihðkÞþabQhðkÞ�1
l¼1ð1þbaiðlÞÞ
; i ¼ 1;2; . . . ;m.
Step 3. Arrange the groups in nondecreasing order of
qðGiÞ ¼rihðkÞ þ a
bQhðkÞ�1l¼1 ð1þ baiðlÞÞ
; i ¼ 1;2; . . . ;m:
Obviously, the complexity of Algorithm 1 is Oðn log nÞ.From Corollary 1, if ðrihðkÞ þ a
bÞQni
l¼hðkÞð1þ baiðlÞÞ 6 ðrjhðkÞ þ abÞQnj
l¼hðkÞð1þ bajðlÞÞ implies thatQni
l¼1ð1þ bailÞPQnj
l¼1ð1þ bajlÞ forall the groups Gi and Gj, then the 1jrij; pij ¼ aijðaþ btÞ; si ¼ s > 0;GTjCmax problem is solvable by Algorithm 1. From Corollary2, the 1jrij; pij ¼ aijðaþ btÞ; si ¼ 0;GTjCmax problem is also solvable by Algorithm 1.
We will illustrate the execution of Algorithm 1 in the following example.
Example 1. In the processing of ingots to prepare them for hot rolling on the blooming mill, the single-machine ready timesscheduling model takes into account the proportional linear deterioration and group technology. Consider 6 ingots (jobs)divided into two groups with: 6 ¼ 8;m ¼ 2; a ¼ 1; b ¼ 0:1; s1 ¼ s2 ¼ 1; t0 ¼ 1, and G1 : fJ1; J2; J3g;a1 ¼ 1;a2 ¼ 2;a3 ¼ 3
2 ; r1 ¼ 5; r2 ¼ 6; r3 ¼ 4; G2 : fJ4; J5; J6g;a4 ¼ 14 ;a5 ¼ 1;a6 ¼ 1
2 ; r4 ¼ 4; r5 ¼ 8; r6 ¼ 2.Solution. According to Algorithm 1, we have:
Step 1: For groups G1 and G2, the optimal job sequences are ½J3; J1; J2� and ½J6; J4; J5�, respectively.Step 2, 3: Next, we compute the following values for groups G1 and G2:G1 : ðr1BðkÞ þ a
bÞQn1
l¼BðkÞð1þ baiðlÞÞ ¼maxfð4þ 10Þð1þ 0:1� 32Þð1þ 0:1� 1Þð1þ 0:1� 2Þ; ð5þ 10Þð1þ 0:1� 1Þð1þ 0:1� 2Þ;
ð6þ 10Þð1þ 0:1� 2Þg ¼ 21:2520, BðkÞ ¼ 1,Qn1
l¼1ð1þ ba1ðlÞÞ ¼ 1:1839, qðG1Þ ¼ 14;
Y.-T. Xu et al. / Applied Mathematical Modelling 38 (2014) 384–391 389
G2 : ðr2BðkÞ þ abÞQn2
l¼BðkÞð1þ baiðlÞÞ ¼maxfð2þ 10Þð1þ 0:1� 12Þð1þ 0:1� 1
4Þð1þ 0:1� 1Þ; ð4þ 10Þð1þ 0:1� 1
4Þð1þ 0:1� 1Þ; ð8þ 10Þð1þ 0:1� 1Þg ¼ 19:8000, BðkÞ ¼ 3,Qn2
l¼1ð1þ ba2ðlÞÞ ¼ 1:7902,qðG2Þ ¼ 8þ10
ð1þ0:1�12Þð1þ0:1�1
4Þ¼ 16:7247.
Since ðr1BðkÞ þ abÞQn1
l¼BðkÞð1þ baiðlÞÞ ¼ 21:2520 > ðr2BðkÞ þ abÞQn2
l¼BðkÞð1þ baiðlÞÞ ¼ 19:8000 andQn1
l¼1ð1þ ba1ðlÞÞ ¼ 1:1839 <Qn2
l¼1
ð1þ ba2ðlÞÞ ¼ 1:7902 are agreeable, from qðG1Þ ¼ 14 < qðG2Þ ¼ 16:7247, Theorems 1 and 2, we know that the optimal groupsequence is ½G1;G2�. Thus, the optimal schedule is ½J3; J1; J2; J6; J4; J5�, and the optimal makespan is 13.1986.
4. A heuristic algorithm and computational experiments for general case
We conjecture that the makespan minimization scheduling problem with ready times, starting-time-dependent processingtimes and independent setup times is NP-hard for the general case. It is easily seen that the optimal job sequence within the samegroup is scheduled in nondecreasing order of rij. From Section 3, we can propose a heuristic algorithm for the general problem.
Algorithm 2.
Step 1. Jobs in each group scheduled in nondecreasing order of rij, i.e.,
rið1Þ 6 rið2Þ 6 � � � 6 riðniÞ; i ¼ 1;2; . . . ;m:
Step 2. Let ðrihðkÞ þ abÞQni
l¼hðkÞð1þ baiðlÞÞ ¼ maxfðrið1Þ þ abÞQni
l¼1ð1þ baiðlÞÞ; ðrið2Þ þ abÞQni
l¼2ð1þ baiðlÞÞ; . . . ; ðriðniÞ þ abÞð1þ baiðniÞÞg;
hðkÞ 2 f1;2; . . . ;nig, calculate hðkÞ and rihðkÞþabQhðkÞ�1
l¼1ð1þbaiðlÞÞ
; i ¼ 1;2; . . . ;m.
Step 3.1. Groups scheduled in nondecreasing order of qðGiÞ ¼rihðkÞþa
bQhðkÞ�1
l¼1ð1þbaiðlÞÞ
;
Step 3.2. Groups scheduled in nondecreasing order of si;Step 3.3. Groups scheduled in nonincreasing order of si
Qnil¼1ð1þ bailÞ.
Step 4. Choose the better solution from Steps 3.1–3.3.
Table 1Results for heuristic algorithm
n m aij 2 U½0:1;1� aij 2 U½0:1;5� aij 2 U½0:1;10�
Mean Max Mean Max Mean Max
20 6 0.0000 0.0001 0.0000 0.0000 0.0000 0.00007 0.0339 1.1695 0.0034 0.1368 0.0000 0.00008 0.0007 0.0272 0.0076 0.3023 0.0000 0.00009 0.0152 0.2729 0.0000 0.0000 0.0000 0.000010 0.0205 0.4514 0.0271 0.5706 0.0069 0.2566
25 6 0.0000 0.0000 0.0000 0.0000 0.0000 0.00007 0.0000 0.0000 0.0000 0.0000 0.0000 0.00008 0.0395 1.1379 0.0015 0.0591 0.0000 0.00009 0.0285 0.3996 0.0000 0.0000 0.0000 0.000010 0.0012 0.0491 0.0050 0.2012 0.0000 0.0000
30 6 0.0000 0.0000 0.0000 0.0000 0.0248 0.99087 0.0000 0.0000 0.0000 0.0000 0.0035 0.14088 0.0122 0.4715 0.0000 0.0000 0.0000 0.00009 0.0000 0.0000 0.0047 0.1867 0.0000 0.000010 0.0053 0.1146 0.0025 0.0990 0.0000 0.0000
35 6 0.0000 0.0000 0.0000 0.0000 0.0000 0.00007 0.0023 0.0692 0.0000 0.0000 0.0000 0.00008 0.0000 0.0000 0.0012 0.0490 0.0000 0.00009 0.0237 0.5255 0.0000 0.0000 0.0000 0.000010 0.0236 0.7221 0.0043 0.1737 0.0000 0.0000
40 6 0.0000 0.0000 0.0000 0.0000 0.0000 0.00007 0.0000 0.0000 0.0000 0.0000 0.0000 0.00008 0.0000 0.0000 0.0000 0.0000 0.0000 0.00009 0.0000 0.0000 0.0000 0.0000 0.0000 0.000010 0.0018 0.0706 0.0000 0.0000 0.0000 0.0000
390 Y.-T. Xu et al. / Applied Mathematical Modelling 38 (2014) 384–391
We conducted computational experiments to evaluate the effectiveness of the heuristic algorithm. We coded the heuristicalgorithm and the branch-and-bound algorithm in VC++ 6.0 and ran the computational experiments on a Pentium 4–2.4 Gbpersonal computer with a RAM size of 2G. For the experiments, the following parameters are considered to generate ran-domly the test problems:
Test problems with a ¼ b ¼ 1, 20, 25, 30, 35 and 40 jobs (n) and with 6, 7, 8, 9 and 10 families (m) (each group must con-tain at least one job) are experimented in our study. The ready times (rij) and the setup times si were generated from theuniform distribution with range [1], respectively. Moreover, to detect the effect of deterioration rate (aij) on the performanceof an algorithm, three different types of deterioration rate ranges were assigned with uniform distribution as follows: [0.1, 1],[0.1, 5] and [0.1, 10].
As a consequence, 75 experimental conditions were examined and 40 replications were randomly generated for each con-dition. A total of 3000 problems were tested. The percentage error of the solution produced by the heuristic algorithm is cal-culated as
ðHA� V�Þ=V� � 100%;
where HA is the makespan of the solution generated by Algorithm 2 and V� is the makespan of the optimal schedule, which isobtained by a complete enumeration (only enumerate groups). The running times for the heuristic algorithm are not given,since most problems are done in no time (a reported CPU time is zero) and the others are finished within a second. The re-sults are summarized in Table 1.
From Table 1, the computational results also show that the proposed Algorithm 2 performs very well in terms of the errorpercentages. The mean error percentage is less than 0.1% for all sizes of problems.
5. Conclusions
A single machine scheduling problem with proportional linear deterioration and group technology assumption had beenstudied. For some special conditions, we showed that the makespan minimization problem with ready times can be solved inpolynomial time. For the general case, we propose a heuristic algorithm. In the future research, it is worthwhile to considerthis open problem (i.e., the computational complexity of the problem remains open), propose more sophisticated and effi-cient heuristic algorithms, or discuss group technology scheduling with learning effect (Jaber [34,35]).
Acknowledgments
The authors are grateful for the editor and two anonymous referees for their helpful comments on earlier version of thearticle. This research was supported by the open project of The State Key Laboratory for Manufacturing Systems Engineering(Grant No. sklms201306), the National Natural Science Foundation of China under Grant No. 11001181, and the Program forLiaoning Excellent Talents in University (Grant No. LJQ2011014).
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