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shbeen studied extensively; see [4,5] for the most recent surveys. Inthis note we consider a particular scheduling problem with con-trollable processing times that can be stated as follows. Supposewe have a set of n independent jobs {1, 2, . . . , n} to be scheduledon a single machine, where job processing times are compressible.Associated with each job j is a normal processing time p0j , a weightwj and a compression rate bj. If job j is compressed by spending anamount uj, then its actual processing time becomes pj = p0j bjuj.Given a budget R, our problem is to find a schedule with minimumwjCj, subject to the constraint that the total amount spent on

compression,uj, does not exceed the fixed budget R. The com-

plexity of this problem is stated as open in [3,4]. In this note weshow that it is NP-hard. Our result implies also that the problemof minimizing

uj, subject to the constraint that

wjCj does not

exceed a given threshold, is also NP-hard.Wan et al. [6] and Hoogeveen and Woeginger [2] studied a

related problem in which the cost functionwjCj + cjuj has

to be minimized. They showed, independently from one another,that their problem is NP-hard [6,2]. On the surface, it may appearthat our problem is somewhat easier to solve than their problem.However, this note shows that our problem is NP-hard as well.

2. Problem reduction

In this section we will show that the decision version of ourproblem is NP-complete. We denote our decision problem as

Corresponding author.E-mail addresses: polytime@cnu.ac.kr (B.-C. Choi), leung@cis.njit.edu

(J.Y.-T. Leung), mpinedo@stern.nyu.edu (M.L. Pinedo).

WCTCC:Given two thresholdsR andWC , a set of n jobs {1, 2, . . . , n}with the initial processing time of job j equal to p0j , the weight ofjob j equal to wj, and the compression rate of job j equal to bj, isthere a schedule such that

wjCj WC and uj R?

We will reduce the Equal Cardinality Partition problem to theWCTCC problem. The Equal Cardinality Partition problem is knownto beNP-complete; see [1]. The Equal Cardinality Partition problemcan be stated as follows.Equal Cardinality Partition: Given 2n integers a1, a2, . . . , a2n suchthat

2nj=1 aj = A, is there a subset I {1, 2, . . . , 2n} such that

|I| = n andjI aj = A2?Given an instance of the Equal Cardinality Partition problem,

we construct an instance of the WCTCC problem as follows. Therewill be 2n jobs. The initial processing time,weight and compressionrate of job j are given as follows.

p0j = B+ aj, wj = B+ aj and bj =B+ ajM + aj , j = 1, . . . , 2n,

where B = nA3 and M = 2n+12n+2B = n(2n+1)2n+2 A3. All jobs can becompletely compressed, i.e., the upper bound of uj is

p0jbj= M +

aj, j = 1, . . . , 2n. The thresholdsWC and R are given asWC = 1

2

((n2 + n)B2 + (n+ 1)AB

+(2nj=1aj

)2+

2nj=1a2j

and R = nM + A2.

Note that since wjp0j= 1, j = 1, . . . , 2n, the jobs that are not

compressed will always follow the compressed jobs in any order.Operations Research Let

Contents lists availa

Operations Re

journal homepage: www

Complexity of a scheduling problem withByung-Cheon Choi a,, Joseph Y.-T. Leung b, Michael La Department of Business, Chungnam National University, 79 Daehakro, Yuseong-gu Daejeb Department of Computer Science, New Jersey Institute of Technology, Newark, NJ 07102,c Department of Information, Operations & Management Sciences, Stern School of Busines

a r t i c l e i n f o

Article history:Received 15 May 2008Accepted 7 October 2009Available online 24 October 2009

Keywords:Total weighted completion timeControllable processing timeSingle machineNP-hard

a b s t r a c t

We consider the problem ofthe total weighted completioor equal to a fixed amount. Twe show that the problem is

1. Introduction

Scheduling problems with controllable processing times have0167-6377/$ see front matter 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.orl.2009.10.011ters 38 (2010) 123126

ble at ScienceDirect

search Letters

.elsevier.com/locate/orl

controllable processing times. Pinedo c

on 305-764, Republic of KoreaUSA, New York University, 44 West 4th Street, New York, NY 10012-1126, USA

scheduling a set of independent jobs on a single machine so as to minimizen time, subject to the constraint that the total compression cost is less thane complexity of this problem is mentioned as an open problem. In this noteNP-hard.

2009 Elsevier B.V. All rights reserved.

the Weighted Completion Time problem with Controllable Cost(WCTCC).

Bthose that are fully compressed, (2) those that are partially com-pressed, and (3) those that are not compressed at all. For jobsin the first category, their actual processing times have been re-duced to zero. Thus, we only need to consider in schedule S jobsin the second and third categories. Without loss of generality, wemay assume that these jobs are scheduled in WSPT order and that2nj=1 uj = R, where WSPT order implies a decreasing order of wjpj .

Clearly, the jobs in the second category are scheduled before the

In this case, we have

B+ akM + ak

B+ alM + al =

(M B)(ak al)(M + ak)(M + al) > 0.

Therefore, we have

n

2A2.

126 B.-C. Choi et al. / Operations Rese

Thus, since A(2nA + 1n2A3 + 1+ 12nA2

)> 0, we have

2nj=1

wjCj > WC + B2 2AB A22B+ A

n2A2

= WC + 2n2A5(A 2nA 1n2A3 1 12nA2 )

2(2B+ A) > WC .In this case, the objective value of the schedule is larger thanWC .Case 3: (|T |, |P|, |U|) = (n 1, 1, n).Let job k be the partially compressed job in S. Clearly, job k isthe first job scheduled in S. Since |T | = n 1 and |P| = 1,jT uj = (n 1)M +

jT aj < R = nM + A2 . Therefore, uk

can be calculated as follows:

uk = RjT(M + aj) = M + A2

jTaj.

Thus, we havejT{k}

uj = R.

The actual processing time of job k can be calculated as follows:

pk = B+ ak B+ akM + ak

(M + A

2jTaj

)

= B+ akM + ak

( jT{k}

aj A2

).

SincejT{k} aj A2 = A2

jU aj, we have

pk = B+ akM + ak

(A2jUaj

).

Since job k is scheduled first in S, we have2nj=1

wjCj =jU

wjCj + pkjU{k}

wj

= 12

(jU(B+ aj)

)2+jU(B+ aj)2

+ pk jU{k}

wj.

Since |U| = n, we have2nj=1

wjCj = 12

((n2 + n)B2 + (2n+ 2)

(jUaj

)B

+(jUaj

)2+jUa2j

+ pk jU{k}

wj.

SincejU aj B22B+ A .

Also, as we showed in Case 2,

> n2A2.

Thus, since A 1 12nA2

> 0 and B = nA3, we have2nj=1

wjCj > WC + B2

2B+ A n2A2 = WC +

2n2A5(A 1 1

2nA2

)2(2B+ A)

> WC .

Again, the objective value of the schedule is larger thanWC .Summarizing the above three cases, the only schedule with

objective value smaller than WC is the one with (|T |, |P|, |U|) =(n, 0, n). All other schedules have objective values larger thanWC .Therefore, if the WCTCC problem has a solution, then the EqualCardinality Partition problem has a solution as well.

By Lemmas 1 and 3, we have the following theorem.

Theorem 1. The WCTCC problem is NP-complete.

Acknowledgements

Byung-Cheon Choi was supported in part by the Korea ResearchFoundation Grant KRF-2008-357-D00289. Joseph Y-T. Leung wassupported in part by the NSF Grant DMI-0556010. Michael L.Pinedo was supported in part by the NSF Grant DMI-0555999.

References

[1] M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theoryof NP-Completeness, W.H. Freeman, New York, 1979.

[2] H. Hoogeveen, G.J. Woeginger, Some comments on sequencing with control-lable processing times, Computing 68 (2002) 181192.

[3] A. Janiak, Scheduling in computer and manufacturing systems, WroclawUniversity of Technology, Wroclaw, Poland, 2006.

[4] A. Janiak, W. Janiak, M. Lichtenstein, Resource management in machinescheduling problems: A survey, DecisionMaking inManufacturing and Services1 (2007) 5989.

[5] D. Shabtay, G. Steiner, A survey of scheduling with controllable processingtimes, Discrete Applied Mathematics 155 (2007) 16431666.

[6] G.Wan, B. Yen, C-L. Li, Singlemachine scheduling tominimize total compressionplus weighted flow cost is NP-hard, Information Processing Letters 79 (2001)273280.

Complexity of a scheduling problem with controllable processing timesIntroductionProblem reductionAcknowledgementsReferences