European Journal of Operational Research 158 (2004) 525–528

www.elsevier.com/locate/dsw

Short Communication

A note on the single machine serial batchingscheduling problem to minimize maximum lateness

with identical processing times

J.J. Yuan a,b, A.F. Yang a,b, T.C.E. Cheng b,*

a Department of Mathematics, Zhengzhou University, Zhengzhou, Henan 450052, People�s Republic of Chinab Department of Management, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People�s Republic of China

Received 29 August 2002; accepted 5 May 2003

Available online 2 August 2003

Abstract

We consider the single machine, serial batching scheduling problem 1jprec; pj ¼ p; s-batch; rjjLmax. The complexity ofthis problem is reported as open in the literature. By reducing this problem to the version without precedence con-

straints, we show that the problem is polynomially solvable.

� 2003 Elsevier B.V. All rights reserved.

Keywords: Scheduling; Precedence constraints; Batches

1. Introduction and problem formulation

Let n jobs J1; J2; . . . ; Jn and a single machinethat can handle only one job at a time be given.

There are precedence relations � between the jobs.Each job Jj has a release date rj, a processing timepj and a weight wj. The jobs are processed inbatches. The completion time of all jobs in a batch

is defined as the finishing time of the last job in the

batch, i.e., the processing time of a batch is equal

to the sum of the processing times of all jobs in the

batch. If a new batch is formed, a constant setup

time s will be incurred. If Ji and Jj are two jobs

* Corresponding author. Tel.: +852-27665215; fax: +852-

23645245.

E-mail address: mscheng@polyu.edu.hk (T.C.E. Cheng).

0377-2217/$ - see front matter � 2003 Elsevier B.V. All rights reserv

doi:10.1016/S0377-2217(03)00361-8

such that Ji � Jj, we only require that Ji is pro-cessed before Jj, and so it is allowed that Ji and Jjare processed in the same batch. Jobs cannot start

before their release dates. We assume that the

starting time of the setup of each batch cannot be

before the maximum release date of the jobs in the

batch. Hence, the starting time of a batch B is atleast maxJj2B rj þ s. Following [3,6], this model iscalled the serial batching scheduling problem and

is denoted by

1jprec; s-batch; rjjf ;

where f is the objective function to be minimized.Coffman et al. [5] provide an algorithm to solve

the problem 1js-batchjP

Cj in Oðn log nÞ time.Albers and Brucker [1] provide an Oðn2Þ timealgorithm to solve the problem 1jprec; pj ¼ p;

ed.

526 J.J. Yuan et al. / European Journal of Operational Research 158 (2004) 525–528

s-batchjP

Cj, and prove that the problems

1js-batchjP

wjCj and 1jchains; pj ¼ 1; s-batchj �PwjCj are unary NP-hard. Webster and Baker [8]

give an algorithm to solve the problem 1js batchjLmax in Oðn2Þ time. Ng et al. [7] reduce the problem1jprec; s-batchjLmax to the problem 1js-batchjLmaxand show that the problem 1jprec; s-batchjLmax canbe solved in Oðn2Þ time by using a revised Web-ster–Baker algorithm. Recently, Baptiste [2] gives

an algorithm to solve the problem 1jpj ¼ p;s-batch; rjjLmax in Oðn14 log nÞ time. Brucker andKnust [4] report that the complexity of the prob-

lem 1jprec; pj ¼ p; s-batch; rjjLmax is still open.We show in this paper that, by appropriately

modifying the release dates and due dates of the

jobs, the problem 1jprec; pj ¼ p; s-batch; rjjLmaxcan be polynomially reduced to the problem

1jpj ¼ p; s-batch; rjjLmax in Oðn2Þ times. Conse-quently, the problem 1jprec; pj ¼ p; s-batch; rjjLmaxis polynomially solvable.

2. Reduction

For the problem 1jprec; pj ¼ p; s-batch; rjjLmax,a feasible schedule (solution) is a pair ðp;BSÞ suchthat p ¼ ðpð1Þ; pð2Þ; . . . ; pðnÞÞ is a sequence of jobindices satisfying the precedence constraints, i.e.,

for any pair of jobs Ji and Jj, if Ji � Jj, thenp1ðiÞ < p1ðjÞ, and BS ¼ fB1;B2; . . . ;Bkg is abatch sequence that partitions the set of all jobs

such that, for each index t with 16 t6 k,fj : JpðjÞ 2 Btg consists of consecutive integers, i.e.,there are two integers x and y with y ¼ xþ jBtj 1such that Bt ¼ fJpðxÞ; Jpðxþ1Þ; . . . ; JpðyÞg. The startingtime of a batch B under a schedule ðp;BSÞ is de-noted by SBðp;BSÞ. The completion time of a jobJj under a schedule ðp;BSÞ is denoted by Cjðp;BSÞ(or simply Cj when there is no ambiguity).

If Ji and Jj are two jobs such that Ji � Jj, thenthe starting time of the batch that contains Jj mustbe at least ri under any feasible schedule. Modi-fying the release date of each job Jj by setting

r0j :¼ maxfrj; rig;

we see that the maximum lateness will not change

under any feasible schedule. Hence, we can re-

cursively modify the release dates of the jobs such

that, for each pair of jobs Ji and Jj, if Ji � Jj, thenr0i 6 r0j. This procedure can be done in Oðn2Þ timeby the standard ‘‘Algorithm Modify rj’’ in thebook by Brucker [3]. The modified release dates

obtained above can be denoted in the followingform:

r0j :¼ maxfri : Ji � Jj or i ¼ jg for 16 j6 n:

Again, if Ji and Jj are two jobs such that Ji � Jj,then the completion time of Ji must be less than orequal to that of Jj under any feasible schedule.Modifying the due date of job Jj by setting

d 0i :¼ minfdj; dig;we see that the maximum lateness will not change

under any feasible schedule. In fact, if ðp;BSÞ is afeasible schedule, thenCiðp;BSÞ6Cjðp;BSÞ, and somaxfCiðp;BSÞ di;Cjðp;BSÞ djg

¼ maxfCiðp;BSÞ d 0i ;Cjðp;BSÞ djg;

i.e., the maximum lateness of Ji and Jj is notchanged after we modify the due date di of Ji to d 0

i .Hence, we can recursively modify the due dates of

the jobs such that, for each pair of jobs Ji and Jj, ifJi � Jj, then d 0

i 6 d 0j. This procedure can be done in

Oðn2Þ time by the standard ‘‘Algorithm Modify

dj’’ in the book by Brucker [3]. The modified duedates obtained above can be denoted in the fol-

lowing form:

d 0i :¼ minfdj : Ji � Jj or i ¼ jg for 16 i6 n:

The above discussion leads to Lemma 2.1:

Lemma 2.1. Any optimal schedule of the problem

1jprec; pj ¼ p; s batch; r0j; d0jjLmax under the modi-

fied release dates and due dates is also an optimal

schedule of the problem 1jprec; pj ¼ p; s batch;rjjLmax under the original release dates and due

dates.

For every job subset J fJ1; J2; . . . ; Jng, a jobJi 2 J is called a minimal job of J if Ji has nopredecessor in J.The general idea of the procedure constructing

a feasible schedule for the problem 1jprec; pj ¼p; s-batch; rjjLmax from an optimal solution for theproblem 1jpj ¼ p; s-batch; r0j; d

0jjLmax can be de-

scribed as follows.

J.J. Yuan et al. / European Journal of Operational Research 158 (2004) 525–528 527

Let ðp;BSÞ be an optimal schedule for the

problem 1jpj ¼ p; s-batch; r0j; d0jjLmax under the

modified release dates and due dates. If p satisfiesthe precedence relations � between the jobs, then

the schedule ðp;BSÞ is also optimal for the versionwith precedence relations, and thus, we havenothing to do. Otherwise, let JpðiÞ be the first job

under p such that JpðjÞ � JpðiÞ for some j > i. LetJp�ðiÞ be the minimal job of the job subset

fJpðjÞ : j > i and JpðjÞ � JpðiÞg:

Since Jp�ðiÞ � JpðiÞ, we must have r0p�ðiÞ 6 r0pðiÞ andd 0

p�ðiÞ 6 d 0pðiÞ. Hence, by exchanging the positions of

jobs JpðiÞ and Jp�ðiÞ in the schedule ðp;BSÞ, thecompletion times of JpðiÞ and Jp�ðiÞ under the new

schedule are not greater than the completion times

of Jp�ðiÞ and JpðiÞ under ðp;BSÞ , respectively, and sothe maximum lateness of the jobs JpðiÞ and Jp�ðiÞ is

not increased under the new schedule. By noting

that the completion time of any job apart from JpðiÞand Jp�ðiÞ is not increased under the new schedule,

the resulting new schedule is still optimal for

1jpj ¼ p; s-batch; r0j; d0jjLmax. This procedure is re-

peated until a schedule ðp�;BS�Þ satisfying theprecedence relations � between the jobs is ob-

tained.

Formally, given any optimal schedule ðp;BSÞfor the problem 1jpj ¼ p; s-batch; r0j; d

0jjLmax, we can

recursively obtain a feasible schedule ðp�;BS�Þ forthe original problem in the following way:

(i) If Jpð1Þ is a minimal job of fJ1; J2; . . . ; Jng, thenset Jp�ð1Þ ¼ Jpð1Þ, and the schedule is not chan-

ged. Otherwise, let Jp�ð1Þ be a minimal job of

J1 ¼ fJj : Jj � Jpð1Þg. Then we exchange thepositions of jobs Jp�ð1Þ and Jpð1Þ in the job se-quence p and batch sequence BS. The presentschedule is denoted by ðp1;BS1Þ.

(ii) Suppose that ðp1;BS1Þ; ðp2;BS2Þ; . . . ; ðpi;BSiÞand also Jp�ð1Þ; Jp�ð2Þ; . . . ; Jp�ðiÞ (16 i6 n 1)have been obtained. If Jpiðiþ1Þ is a minimal

job of fJpiðiþ1Þ; Jpiðiþ2Þ; . . . ; JpiðnÞg, then set

Jp�ðiþ1Þ ¼ Jpiðiþ1Þ, and the schedule is not chan-

ged. Otherwise, let Jp�ðiþ1Þ be a minimal job ofJiþ1 ¼ fJpiðjÞ : j > iþ 1 and JpiðjÞ � Jpiðiþ1Þg.Then we exchange the positions of jobs

Jp�ðiþ1Þ and Jpiðiþ1Þ in the job sequence pi and

batch sequence BSi. The present schedule is

denoted by ðpiþ1;BSiþ1Þ.(iii) Set ðp�;BS�Þ ¼ ðpn;BSnÞ:

Clearly, the creation of the schedule ðp�;BS�Þtakes at most Oðn2Þ time. By the definition ofðp�;BS�Þ and the above discussion, we can easilysee the correctness of the following lemma.

Lemma 2.2. ðp�;BS�Þ is an optimal schedule for the

modified problem

1jpj ¼ p; s batch; r0j; d0jjLmax

under the modified release dates and due dates.

Furthermore, if Jp�ðiÞ and Jp�ðjÞ are two jobs such that

Jp�ðiÞ � Jp�ðjÞ, then i < j.

Combining Lemmas 2.1 and 2.2, we deduce that

ðp�;BS�Þ is an optimal schedule for the originalproblem 1jprec; pj ¼ p; s-batch; rjjLmax. Hence,

the problem 1jprec; pj ¼ p; s-batch; rjjLmax can bepolynomially reduced to the problem 1jpj ¼ p;s-batch; rjjLmax in Oðn2Þ time.Recall that Baptiste [2] gives an Oðn14 log nÞ

algorithm for the problem

1jpj ¼ p; s-batch; rjjLmax:This implies that the problem 1jprec; pj ¼ p;s-batch; rjjLmax can be solved in Oðn14 log nÞ time aswell.

3. Conclusions

In this paper, we have studied the single

machine, serial batching scheduling problem

1jprec; pj ¼ p; s-batch; rjjLmax. The complexity ofthis problem is reported as open in the literature.

By reducing this problem to the version without

precedence constraints, we show that the problem

is solvable in Oðn14 log nÞ time. Since the highorder of the polynomial is not practical, any

improvement in the complexity is interesting.

Acknowledgements

This research was supported in part by The

Hong Kong Polytechnic University under the

528 J.J. Yuan et al. / European Journal of Operational Research 158 (2004) 525–528

ASD in China Business Services. The first author

was also supported in part by the National Nat-

ural Science Foundation of China under the grant

number 10371112.

References

[1] S. Albers, P. Brucker, The complexity of one-machine

batching problems, Discrete Applied Mathematics 47 (1993)

87–107.

[2] P. Baptiste, Batching identical jobs, Mathematical Methods

of Operations Research 52 (2000) 355–367.

[3] P. Brucker, Scheduling Algorithms, Springer Verlag, Berlin,

2001.

[4] P. Brucker, S. Knust, Complexity results for scheduling

problems, http://www.mathematik.uni-osnabrueck.de/re-

search/OR/class/, 2002.

[5] E.G. Coffman, M. Yannakakis, M.J. Magazine, C. Santos,

Batch sizing and sequencing on a single machine, Annals of

Operations Research 26 (1990) 135–147.

[6] J.K. Lenstra, A.H.G. Rinnooy Kan, P. Brucker, Complexity

of machine scheduling problems, Annals of Discrete Math-

ematics 1 (1977) 343–362.

[7] C.T. Ng, T.C.E. Cheng, J.J. Yuan, A note on the single

machine serial batching scheduling problem to minimize

maximum lateness with precedence constraints, Operations

Research Letters 30 (2002) 66–68.

[8] S.T. Webster, K.R. Baker, Scheduling groups of jobs on

a single machine, Operations Research 43 (1995) 692–

703.