2006-2007 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptPublishe

Cheng, TCE

Ng, CTD

Yuan, JJ

European Journal of Operational Research

2006-2007 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptPublishedGreen (AAM

PolyU Institutional Repository

NP-hardness of the single-variable-resourcescheduling problem to minimize the total weightedcompletion timeJ.J. Yuan1,2, T.C.E. Cheng2∗, and C.T. Ng21Department of Mathematics, Zhengzhou University,Zhengzhou, Henan 450052, People’s Republic of China2Department of Logistics, The Hong Kong Polytechnic University,Hung Hom, Kowloon, Hong Kong, People’s Republic of ChinaAbstract: Baker and Nuttle [1] studied the following single-variable-resourcescheduling problem: sequencing n jobs for processing by a single resource to minimizea function of job completion times, when the availability of the resource varies over time.When the objective function to be minimized is the total weighted completion time, Bakerand Nuttle conjectured that the problem is NP-hard. We show in this note that the con-jecture is true.1 IntroductionConsider the problem of sequencing a set N = {1, 2, ..., n} of jobs to be processed usinga single resource, where the availability of the resource varies over time. This problem,known as single-variable-resource scheduling, was studied in Baker and Nuttle [1].For a time instant t ≥ 0, we denote by r(t) the resource available at time t. For eachj with 1 ≤ j ≤ n, we denote by pj > 0 the resource requirement of job j. Once pj unitsof resource have been applied to job j, the job is considered complete. We assume thatall the jobs are released at time 0, the processing of each pair of distinct jobs does notoverlap, the processing of every job is continuous (i.e., no preemption is allowed), andthere is no idle time between the processing of consecutive jobs. So, a schedule pi of thejobs can be specified by a sequencepi = (pi(1), pi(2), ..., pi(n)),∗Corresponding author1This is the Pre-Published Version.where pi(i) is the job in the i-th position of pi. It was shown in [1] that, under schedule pi,the completion time of job pi(j) can be expressed as a function of the formCpi(j)(pi) = f(ppi(1) + ppi(2) + ...+ ppi(j)).For the purpose of this paper, we only consider Cpi(j) in the case that there are positivenumbers X, a and b such thatr(t) = a, if 0 ≤ t < X,b, if t ≥ X.In this case, the completion time of job pi(j) can be determined by the functionCpi(j) = Ppi(j)/a, if Ppi(j) ≤ aX,X + (Ppi(j) − aX)/b, if Ppi(j) > aX,wherePpi(j) = ppi(1) + ppi(2) + ...+ ppi(j).For the above problem, when the objective function to be minimized is the totalweighted completion time, i.e.,∑1≤j≤nwjCj(pi), Baker and Nuttle [1] conjectured thatthe problem, denoted by 1∗||∑wjCj, is NP-hard. It was hinted in Baker and Smith [2]that the computational complexity of 1∗||∑wjCj is still open up to now.We show in this note that Baker and Nuttle’s conjecture is true, i.e., 1∗||∑wjCj isNP-hard. All the numbers appearing in this note are assumed to be non-negative andintegral. So, the result of this note still holds under the variable assumptions in [1].2 NP-hardness ProofThe standard single-machine total weighted tardiness problem is denoted by 1||∑wjTj,where each job j has a due date dj, and under a schedule pi, the tardiness of job j isdefined byTj(pi) = max{0, Cj(pi)− dj}.The decision version of 1||∑wjTj is denoted by 1||∑wjTj ≤ Y .By Yuan [3], the scheduling problem 1|dj = d|∑wjTj is NP-hard. In fact, it wasshown in [3] that the decision version 1|dj = d|∑wjTj ≤ Y is NP-complete.Theorem 1∗||∑wjCj is NP-hard.Proof Let us be given an instance I of 1|dj = d|∑wjTj ≤ Y :(p1, , ..., pn;w1, ..., wn; d;Y ),2where we have n jobs J1, ..., Jn; pj ≥ 1 is the processing time of job Jj, 1 ≤ j ≤ n; wj ≥ 1is the weight of job Jj, 1 ≤ j ≤ n; d < p1+p2+ ...+pn is the common due date of the jobs;and Y > 0 is the threshold value for the total weighted tardiness of the jobs. The decisionproblem asks if there is a schedule σ for the n jobs such that∑1≤j≤nwjTj(σ) ≤ Y .We first define ∆ = (w1 + w2 + ... + wn)(p1 + p2 + ... + pn). Then we construct aninstance I∗ of the single-variable-resource scheduling problem 1∗||∑wjCj ≤ Q as follows.• n jobs J ′1, J ′2, ..., J ′n with J ′j, 1 ≤ j ≤ n, corresponding to job Jj in I.• The resource requirement of job J ′j is defined by p′j = 2∆pj, 1 ≤ j ≤ n, and the weightof job J ′j is defined as w′j = wj, 1 ≤ j ≤ n.• The resource availability function is defined asr(t) = 2∆, if 0 ≤ t ≤ d,1, if t > d.• Threshold value is defined as Q = ∆(2Y + 1).• The decision problem asks if there is a schedule pi such that∑1≤j≤nw′jC ′j(pi) ≤ Q, whereC ′j(pi) is the completion time of job J′j under pi.For any permutation h of (1, 2, ..., n), h can be thought of as a schedule for both I andI∗. By the definition of r(t), it can be observed that, under schedule h, the completiontime of job J ′j isC ′j(h) = Cj(h), if Cj(h) ≤ d,d+ 2∆Tj(h), if Cj(h) > d,and so,2∆Tj(h) ≤ C ′j(h) ≤ Cj(h) + 2∆Tj(h).By noting that ∆ ≥ ∑1≤j≤nwjCj(h), one can observe that(1) :∑1≤j≤nw′jC′j(h) ≤ ∆+ 2∆∑1≤j≤nwjTj(h)and(2) : 2∆∑1≤j≤nwjTj(h) ≤∑1≤j≤nw′jC′j(h).From (1), if∑1≤j≤nwjTj(h) ≤ Y , then we must have∑1≤j≤nw′jC′j(h) ≤ ∆+ 2∆Y = Q.From (2), if∑1≤j≤nw′jC′j(h) ≤ Q = ∆(2Y + 1), we must have∑1≤j≤nwjTj(h) ≤ Y + 1/2,3and by the integrality of the data, we further have∑1≤j≤nwjTj(h) ≤ Y.It follows that instance I has a feasible solution if and only if instance I∗ has a feasiblesolution. The result follows. 2Remark: By [3], 1|dj = d|∑wjTj is NP-hard in the ordinary sense. So, we haveproved that 1∗||∑wjCj is NP-hard. It is still open if the problem 1∗||∑wjCj is NP-hardin the strong sense. The method presented in this note does not work for the strongNP-hardness of 1∗||∑wjCj.AcknowledgementsThis research was supported in part by The Hong Kong Polytechnic University undergrant number S818. The first author was also supported in part by the National NaturalScience Foundation of China under grant number 10371112.References[1] K.R. Baker and H.L.W. Nuttle, Sequencing independent jobs with a single resource,Naval Research Logistics Quarterly, 27(1980), 499-510.[2] K.R. Baker and J.C. Smith, A multiple-criterion model for machine scheduling. Jour-nal of Scheduling, 6(2003), 7-16.[3] J.J. Yuan, The NP-hardness of the single machine common due date weighted tar-diness problem, Systems Science and Mathematical Sciences, 5(1992), 328-333.4

NP-hardness of the single-variable-resource scheduling problem to minimize the total weighted completion time

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