Applied Mathematics and Computation 242 (2014) 309–314

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Applied Mathematics and Computation

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

Single-machine bicriterion group scheduling with deterioratingsetup times and job processing times

http://dx.doi.org/10.1016/j.amc.2014.05.0480096-3003/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author.E-mail address: wangjianjun_2008@163.com (J.-J. Wang).

Jian-Jun Wang ⇑, Ya-Jing LiuFaculty of Management and Economics, Dalian University of Technology, Dalian 116024, PR China

a r t i c l e i n f o

Keywords:Single-machineGroup schedulingBicriterion schedulingDeteriorating jobs

a b s t r a c t

This paper considers a group scheduling problem with two ordered criteria where bothsetup times and job-processing times are increasing functions of their starting times. Itis assumed that the jobs be classified into several groups and the jobs of the same grouphave to be processed contiguously. We consider two objectives where the primary criterionis the total weighted completion time and the secondary criterion is the maximum cost. Apolynomial time algorithm is presented to solve this bicriterion group scheduling problemwith deteriorating setup times and job-processing times. This algorithm can also solve sin-gle-machine group scheduling problems with deteriorating setup times and job-processingtimes in several ordered maximum cost and arbitrary precedence.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

In classical group scheduling problems, most research assumes that the processing time of a job and the setup time of agroup be a constant. However, there are many situations in which a job or a group that is processed later consumes moretime than the same job or group when processed earlier. For example, in fire fighting when the time and effort requiredto control a fire increases if there is a delay in the start of the fire-fighting effort. Hence, there is a growing interest in theliterature to study scheduling problems involving deteriorating jobs, i.e. jobs whose processing times are increasing func-tions of their starting times. We refer the reader to the book by Gawiejnowicz [1] for more details on single-machine,parallel-machine and dedicated-machine scheduling problems with deteriorating jobs.

On the other hand, scheduling problems with group technology have been studied by Baker [2], Cheng et al. [3], Ham et al.[4], Mitrofanov [5], Ozden et al. [6], Webster and Baker [7], to name a few. To the best of our knowledge, only a few resultsconcerning scheduling problems with deteriorating jobs and group technology simultaneously are known. Since longer setupor preparation might be necessary as food quality deteriorates or a patient’s condition worsens, Wu et al. [8] considered a sit-uation where the group setup times and job processing times are both described by a simple linear deterioration function.They showed that the makespan and the total completion time problems remain polynomially solvable under the proposedmodel. Wang et al. [9] considered the same model with Wu et al. [8], but with proportional deterioration. They proved that themakespan minimization problem and total weighted completion time minimization problem can be solved in polynomialtime. Wang et al. [10] considered the same model with Wu et al. [8], but with a more general linear deterioration. For singlemachine group scheduling, they proved that the makespan minimization problem can be solved in polynomial time. Wanget al. [11] considered a single machine scheduling problem with deteriorating jobs, ready times and group technology, in

310 J.-J. Wang, Y.-J. Liu / Applied Mathematics and Computation 242 (2014) 309–314

which the group setup times are assumed to be known and fixed. For a special case, they showed that the makespan minimi-zation problem can be polynomially solvable. More recent papers which have considered scheduling jobs with deterioratingjobs and group technology include Yang [12], Yang and Yang [13], Wei and Wang [14], Cheng et al. [15], Bai et al. [16], Lee andLu [17], Huang and Wang [18], Wang et al. [19], Xu et al. [20], Lu et al. [21], Wang and Wang [22], and Yin et al. [23].

However, traditional research on the scheduling problem with deteriorating jobs assume that all jobs to be processedhave a single criterion. In real production settings and service environments, scheduling decisions are made with respectto bicriterion (multicriteria) performance rather than a single criterion (T’kindt and Billaut [24]). In this paper we considera bicriterion scheduling problem with deteriorating setup times and deteriorating job processing times. This bicriterionmodel was proposed by Cheng et al. [3]. The remainder of this paper is organized as follows. In Section 2 we provide thenotation and formulation of the problem. Section 3 deals with the makespan minimization problem. Concluding remarksare given in the last section.

2. Notation and problem statement

In this section, the notation that is used throughout the paper will be introduced first, followed by the formulation of theproblem.

Notation.

G

the number of groups ðG P 2Þ Gi group i; i ¼ 1;2; . . . ;G ni the number of jobs in Gi; i ¼ 1;2; . . . ;G n the total number of jobs i.e., ðn1 þ n2 þ . . .þ nG ¼ nÞ Jij job j in Gi; i ¼ 1;2; . . . ;G; j ¼ 1;2; . . . ;ni

di

the deterioration rate of setup time for Gi

aij

the deterioration rate for the jth job in Gi

si

the actual setup time of Gi

pij

the actual processing time of Jij

xij

the relative importance for the jth job in Gi

Cij

the completion time for the jth job in Gi

CijðpÞ

the completion time of Jij under schedule p

There are n independent non-preemptive jobs to be scheduled for processing on a single machine. It is assumed that alljobs are available at time t0, where t0 > 0, without idle time and at most one job at a time. All jobs are classified into G 6 ngroups. Jobs in the same group have to be processed contiguously. A setup time precedes the processing of each group. Theactual job-processing time of Jij is a linear function of its starting time t, that is,

pij ¼ aijt; i ¼ 1;2; . . . ;G; j ¼ 1;2; . . . ;ni:

Moreover, the actual setup time of Gi is also a linear function of its starting time t and as follow:

si ¼ dit; i ¼ 1;2; . . . ;G:

Given a schedule, the completion time Cij for each job j in group i is easily determined. The quality of a schedule ismeasured by two criteria: the total weighted completion time

PPxijCij ¼

PGi¼1

Pnij¼1xijCij and the maximum cost

fmax ¼max fijðCijÞji ¼ 1;2; . . . ;G; j ¼ 1;2; . . . ;ni� �

, where all cost functions fij are nondecreasing in the job completion times.It is given that the primary criterion is the total weighted completion time and the secondary criteria is the maximum cost.Our primary criterion is one of the most important scheduling criteria. The objective is to minimize fmax on the set of schedulesminimizing

PPxijCij. Let GT denote the group scheduling problem. Using the conventional notation (Graham et al. [25]), the

problem of minimizing fmax on the set of schedules minimizingPP

xijCij is denoted as 1jGT; si ¼ dit; pij ¼aijtj

PPxijCij; f max

� �.

3. Transformation of problem and the polynomial time algorithm

3.1. Formulating the bicriterion problem as a single criterion problem

We introduce precedence constraints on a set of groups and on the set of jobs for each group and formulate our bicriterionproblem as a single criterion problem to minimize fmax subject to these precedence constraints. First of all, we give two lemmas.

Lemma 1. If the sequence of groups is fixed, then a schedule minimizes the total weighted completion time is obtained whenaij

xijð1þaijÞ of each job in Gi ði ¼ 1;2; . . . ;GÞ is increasing.

J.-J. Wang, Y.-J. Liu / Applied Mathematics and Computation 242 (2014) 309–314 311

Proof. Let the sequence of groups be fixed. Consider an optimal schedule p in Gi. Suppose there are two adjacent jobs Jik andJij in p such that Jik immediately precedes Jij and aik

xikð1þaikÞ>

aij

xijð1þaijÞ. Let the starting time of Jik be T0. Suppose C denote the sum

of weighted completion time of the remaining jobs in Gi. We have

CikðpÞ ¼ T0ð1þ aikÞCijðpÞ ¼ T0ð1þ aikÞð1þ aijÞXni

j¼1

xijCijðpÞ ¼ C þxikT0ð1þ aikÞ þxijT0ð1þ aikÞð1þ aijÞ:

Performing an adjacent pair-wise interchange of Jij and Jik to get a new schedule p0, we have

Cijðp0Þ ¼ T0ð1þ aijÞCikðp0Þ ¼ T0ð1þ aikÞð1þ aijÞXni

j¼1

xijCijðp0Þ ¼ C þxijT0ð1þ aijÞ þxikT0ð1þ aikÞð1þ aijÞ

Xni

j¼1

xijCijðp0Þ �Xni

j¼1

xijCijðpÞ ¼ xikT0ð1þ aikÞaij �xijT0ð1þ aijÞaik:

Since aikxikð1þaikÞ

>aij

xijð1þaijÞ, then

Pnij¼1xijCijðp0Þ �

Pnij¼1xijCijðpÞ < 0, Hence,the value of the objective function under p0 is

strictly less than that under p. This contradicts the optimality of p and proves the lemma. h

Lemma 2. If the sequence of jobs in each group are fixed, then a schedule minimizes the total weighted completion time is obtained

according to increasing of qi of each group, where qi ¼ð1þdiÞ

Qnij¼1ð1þaijÞ�1

ð1þdiÞPni

j¼1xijP

jk¼1ð1þaikÞ

; i ¼ 1;2; . . . ;G.

Proof. Let the job sequence for each group be fixed. Consider an optimal schedule S where groups do not follow the increas-ing of qi. In this schedule, suppose there are two adjacent group GA and GB in S such that GA immediately precedes GB andqA > qB. We assume that the starting time of the group GA be T. We now perform an adjacent pairwise interchange of GA

and GB, leaving the remaining groups in their original positions, to derive a new sequence S0. Suppose H denotes the sumof weighted completion time of the remaining group. Under S and S0, we have

XG

i¼1

Xni

j¼1

xijCijðSÞ ¼ H þ Tð1þ dAÞXnA

j¼1

xAj

Yj

k¼1

ð1þ aAkÞ !

þ Tð1þ dAÞð1þ dBÞYnA

k¼1

ð1þ aAkÞXnB

j¼1

xBj

Yj

k¼1

ð1þ aBkÞ !

:

XG

i¼1

Xni

j¼1

xijCijðS0Þ ¼ H þ Tð1þ dBÞXnB

j¼1

xBj

Yj

k¼1

ð1þ aBkÞ !

þ Tð1þ dBÞð1þ dAÞYnB

k¼1

ð1þ aBkÞXnA

j¼1

xAj

Yj

k¼1

ð1þ aAkÞ !

:

In addition, we have

XG

i¼1

Xni

j¼1

xijCijðS0Þ �XG

i¼1

Xni

j¼1

xijCijðSÞ ¼ ð1þ dBÞYnB

k¼1

ð1þ aBjÞ � 1

" #ð1þ dAÞT

XnA

j¼1

xAj

Yj

k¼1

ð1þ aAkÞ ! !

� ð1þ dAÞYnA

k¼1

ð1þ aAjÞ � 1

" #ð1þ dBÞT

XnB

j¼1

xBj

Yj

k¼1

ð1þ aBkÞ ! !

:

Q Q

Since qA > qB, i.e.,

ð1þdAÞnAj¼1ð1þaAjÞ�1

ð1þdAÞPnA

j¼1xAjP

jk¼1ð1þaAkÞð Þ >

ð1þdBÞnBj¼1ð1þaBjÞ�1

ð1þdBÞPnB

j¼1xBjP

jk¼1ð1þaBkÞð Þ, hence, we have

PGi¼1

Pnij¼1xijCijðS0Þ �

PGi¼1

Pnij¼1xijCijðSÞ

< 0. This contradicts the optimality of S and proves the lemma. h

Definition 1. For any pair of jobs Jik and Jij from the same group Gi; Jik ! Jij if and only if aikxikð1þaikÞ

<aij

xijð1þaijÞ.

Definition 2. For any pair of groups GA; GB; GA ) GB if and only if

ð1þ dAÞQnA

j¼1ð1þ aAjÞ � 1

ð1þ dAÞPnA

j¼1 xAjPjk¼1ð1þ aAkÞ

� � < ð1þ dBÞQnB

j¼1ð1þ aBjÞ � 1

ð1þ dBÞPnB

j¼1 xBjPjk¼1ð1þ aBkÞ

� � :

312 J.-J. Wang, Y.-J. Liu / Applied Mathematics and Computation 242 (2014) 309–314

According to Lemma 1, a schedule is feasible with precedence relation! on the set of jobs for each group. And Lemma 2give that a schedule is feasible with precedence relation ) on the set of groups. Therefore, we formulate our bicreterionproblem as a single criterion problem to minimize fmax subject to two precedence constraints ! and ). Then we havethe following theorem.

Theorem 1. The problem 1jGT; si ¼ dit; pij ¼ aijtjPP

xijCij; f max� �

is equivalent to the problem 1jGT; si ¼ dit; pij ¼ aijt; !;) jfmax.

Proof. Lemmas 1 and 2 show that any schedule minimizingPP

xijCij is feasible with respect to the relations! and) andvice versa. So the problems 1jGT; si ¼ dit; pij ¼ aijtj

PPxijCij; f max

� �and 1jGT; si ¼ dit; pij ¼ aijt; !; ) jfmax are equiva-

lent. h

3.2. A polynomial time algorithm

In this section we give an algorithm to solve the problem 1jGT; si ¼ dit; pij ¼ aijt; !; ) jfmax. The algorithm is amodification of the Lawler’s algorithm [26] presented for solving the single machine problem of minimizing maximum costsubject to precedence constraint.

Firstly,we consider the scheduling of jobs in any group. We assume U be makespan. We consider the problem ofscheduling jobs of Gi to minimize fmax subject to the relation! and condition that all jobs are available for processing at time

Uð1þdiÞ

Qnij¼1ð1þaijÞ

. Let pGidenote an optimal sequence of jobs for Gi problem. Let FGi

ðUÞ ¼maxj¼1;2;...;nifijðCijÞ� �

. Then, we give the

algorithm of jobs in Gi as follow,

Lawler’s Algorithm

Step 1. Compute U ¼QG

k¼1 ð1þ dkÞQnk

j¼1ð1þ akjÞ� �

, set T ¼ U and S ¼ fJijjj ¼ 1;2; . . . nig.Step 2. Determine the set L containing the jobs that have no successors in S with respect to the relation !.Step 3. Choose Jij 2 L that has the minimal fijðTÞ value, breaking ties arbitrarily. Jij is processed from T

ð1þaijÞto time T.

Step 4. Set T ¼ Tð1þaijÞ

and S ¼ S� fJijg. If S – 0, then go to Step 2; otherwise, an optimal sequence pGiis constructed.Calculate

FGiðUÞ and stop.

Secondly, we consider the scheduling of groups. we let M be the set of groups that have no successors with respect to therelation ). Let FGk

ðUÞ ¼minGi2MfFGiðUÞg, then there exists an optimal schedule for the problem

1jGT; si ¼ dit; pij ¼ aijt; !; ) jfmax in which jobs of Gk are sequenced last in the order determined by pGk. Then we give

the algorithm of groups as follow,

Modified Lawler’s algorithm

Step 1. Set R ¼ f1;2; . . . ;Gg.Step 2. Determine the set M containing the groups that have no successors in R with respect to the relation ).Step 3. Choose Gk 2 M that has the minimal FGk

ðUÞ value, breaking ties arbitrarily. Jobs of Gk are processed from timeU

ð1þdkÞQnk

j¼1ð1þakjÞ

to time U in the pGkorder.

Step 4. Set U ¼ Uð1þdkÞ

Qnkj¼1ð1þakjÞ

and R ¼ R� fGkg. If R – 0, then go to Step 2; otherwise, an optimal group schedule is

constructed. Calculate the corresponding optimal values of fmax andPP

xijCij and stop.

Theorem 2. The Modified Lawler’s algorithm solves the problem 1jGT; si ¼ dit; pij ¼ aijt; !; ) jfmax in OðGn2Þ time.

Proof. Let R ¼ f1;2; . . . ;Gg be the set of all groups, and let M # R be the set of groups without successors. For any subsetQ # R, let f �maxðQÞ denote the optimal values of the criterion fmax for the problem in which Q is the set of all groups. Clearly,f �maxðRÞ satisfies the following inequalities:

f �maxðRÞP minGi2MfFGiðUÞg

f �maxðRÞP f �maxðR� figÞ for all i 2 R:

Then, let Gk 2 M be such a group that FGkðUÞ ¼ minGi2MfFGi

ðUÞg. We have

f �maxðRÞP max FGkðUÞ; f �maxðR� fkgÞ

� �:

J.-J. Wang, Y.-J. Liu / Applied Mathematics and Computation 242 (2014) 309–314 313

The right hand side of this inequality is precisely the cost of an optimal schedule subject to the condition that jobs of Gk

are processed last in pGkorder. So there exists an optimal schedule in which jobs of Gk are processed last in the pGk

order.Since Gk is the group that is selected by the Modified Lawler’s algorithm, repeating the above argument shows that the Mod-ified Lawler’s algorithm solves the problem 1jGT; si ¼ dit; pij ¼ aijt; !; ) jfmax. The same argument can be applied to showthat Lawler’s algorithm solves the problem of Gi for each i. So the algorithm of the Modified Lawler’s algorithm is correct.Now, let we establish the time complexity of the Modified Lawler’s algorithm. Step 1,2 and 4 require Oðnþ G2Þ time. Step

3 require O GPG

i¼1jij2

� �, or equivalently, OðGn2Þ. Therefore, the overall time complexity is OðGn2Þ. h

That is, the original bicriterion problem 1jGT; si ¼ dit; pij ¼ aijtjPP

xijCij; f max

� �is solved in OðGn2Þ time.

To demonstrate our method, we consider the problem 1jGT; si ¼ dit; pij ¼ aijtjPP

xijCij; Lmax� �

, wherefijðCijÞ ¼ Lmax ¼maxfCij � dijg is the secondary criterion, dij is due date of the job Jij in group Gi. Assume we have six jobs,the deterioration rate of each job, the deterioration rate of setup time, weights and due dates are given as follows:

J

1 2 3 4 5 6

aj

0.3 0.5 0.125 1 0.5 1 xj 1 2 2 1 3 1 dj 20 18 15 40 20 17

We assume that all jobs be classified into three group, say GA; GB and GC and setup times be dA ¼ 14 ; dB ¼ 1; dC ¼ 1

4. Andthere are GA ¼ f1g ¼ fJA1g; GB ¼ f2;3g ¼ fJB1; JB2g and GC ¼ f4;5;6g ¼ fJC1; JC2; JC3g. Suppose t0 ¼ 1.

We first respectively calculate aij

xijð1þaijÞin group GB and GC . we have aB1

xB1ð1þaB1Þ¼ 1

6 ;aB2

xB2ð1þaB2Þ¼ 1

18 ;aC1

xC1ð1þaC1Þ¼ 1

2 ;aC2

xC2ð1þaC2Þ¼ 1

9 ;aC3

xC3ð1þaC3Þ¼ 1

2. So precedence relation! in group GB is JB2 ! JB1. Precedence relation

! in group GC is JC2 ! JC3 and JC2 ! JC1. Apply Lawler’s algorithm to solve problem of group GC , we calculateT ¼ U ¼ t0ð1þ dAÞð1þ dBÞð1þ dCÞð1þ aA1Þð1þ aB1Þð1þ aB2Þð1þ aC1Þð1þ aC2Þð1þ aC3Þ ¼ 41:16, and S ¼ fJC1; JC3g, we canobtain fC1ðTÞ ¼ 1:16; f C3ðTÞ ¼ 24:16. So precedence is JC2 ! JC3 ! JC1. Then we calculate qi of each group

qi ¼ð1þdiÞ

Qnij¼1ð1þaijÞ�1

ð1þdiÞPni

j¼1xijP

jk¼1ð1þaikÞð Þ

� . qA ¼ qC ¼ 0:39; qB ¼ 0:21, so precedence relations of all groups is GB ) GA and GB ) GC . Apply

Modified Lawler’s algorithm to solve the problem 1jGT; si ¼ dit; pij ¼ aijt; !; ) jfmax. We obtain M ¼ fGA;GCg and calculateFGAðUÞ ¼ 21:16; FGC ðUÞ ¼ 3:58. That is FGA

ðUÞ > FGC ðUÞ, then we have GC are processed last. Therefore, the final optimalschedule is ðJB2; JB1; JA1; JC2; JC3; JC1Þ i.e. ð3;2;1;5;6;4Þ with value CA1 ¼ 5:49; CB1 ¼ 3:38; CB2 ¼2:25; CC1 ¼ 41:16; CC2 ¼ 10:29; CC3 ¼ 20:58 and

PPxijCij ¼ 109:36; Lmax ¼ 3:58.

4. Conclusions

In this paper we considered single-machine bicriterion group scheduling problems with deteriorating setup times anddeteriorating job processing times. We transform the primary criterion which the total weighted completion time is minimalto two precedence constraints condition. Then we formulate the bicriterion problem as a single criterion problem tominimize fmax subject to two specially constructed precedence constraints. An effective algorithm is given in this paper.Certainly, the algorithm can also be adopted to solve the problem with multiple maximum cost criteria,1jGT; si ¼ dit; pij ¼ aijt; !; ) jðg1; g2; . . . ; gmÞ in OðGmn2Þ, where criteria gk are numbered in decreasing order of theirrelative importance. For further research, it would be interesting to extend the results to the case with multiple machines.Moreover, one may consider more general nonlinear deterioration types.

Acknowledgements

The authors are grateful for two anonymous referees for their helpful comments on earlier version of the article. Thisresearch was supported by the National Natural Science Foundation of China (Grant Nos. 11001181 and 71271039), NewCentury Excellent Talents in University (NCET-13-0082), Changjiang Scholars and Innovative Research Team in University(IRT1214), the Fundamental Research Funds for the Central Universities (DUT14YQ211).

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