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Operations Research Letters 31 (2003) 49–52

OperationsResearchLetters

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An improved max-#ow-based lower bound for minimizingmaximum lateness on identical parallel machines

Mohamed Haouari∗, Anis GharbiLaboratory of Mathematical Engineering, Ecole Polytechnique de Tunisie, BP 743, 2078 La Marsa, Tunisia

Received 30 January 2002; received in revised form 13 June 2002; accepted 18 June 2002

Abstract

In this paper, we introduce a new concept of semi-preemptive scheduling and we show how it can be used to derive amaximum-#ow-based lower bound for the P|rj|Lmax which dominates the well-known preemptive lower bound. We showthat, in some cases, the proposed bound strictly dominates the preemptive one while having the same complexity.c© 2002 Elsevier Science B.V. All rights reserved.

Keywords: Identical parallel machines; Release dates; Due dates; Maximum lateness; Lower bound; Preemption

1. Introduction

In this paper, we propose a new strong lower boundfor minimizing the maximum lateness on a set of iden-tical parallel machines subject to release dates. Thisscheduling problem, which is denoted by P|rj|Lmax, isformally de<ned as follows: a set J of n jobs has tobe scheduled on m identical parallel machines (withn¿m ¿ 2). Each job j has a processing time pj, arelease date rj, and a due date dj. Each job is con-strained to start after its release date, and should beideally completed before its due date. All data are as-sumed to be deterministic and integer. Each machineprocesses at most one job at one time and each jobcannot be processed by more than one machine at onetime. We assume that preemption is not allowed, andthat all machines are ready from time zero onwards.The lateness of a job j is de<ned as Lj=Cj−dj, whereCj denotes its completion time. The objective is to

∗ Corresponding author. Fax +2161-748-843.E-mail address: mohamed.haouari@ept.rnu.tn (M. Haouari).

<nd a schedule that minimizes the maximum latenessLmax = maxj∈J Lj.It is worth noting that setting a delivery time qj =

D−dj for all j ∈ J (where D=maxj∈J dj), permits torestate the P|rj|Lmax as an equivalent P|rj; qj|Cmax (i.e.minimizing the maximum completion time on identi-cal parallel machines with release dates and deliverytimes). For convenience, in the sequel, we refer in-variably to any of these two equivalent forms.The P|rj|Lmax isNP-hard in the strong sense since

it generalizes the well-studied 1|rj|Lmax which isstronglyNP-hard [3]. However, the optimal Lmax ofits preemptive relaxation, denoted by P|rj; pmtn|Lmax,can be computed in polynomial time [5]. Hence,it provides a strong lower bound for the P|rj|Lmax,denoted hereafter by the preemptive lower bound(PLB). Indeed, although many other lower boundshave been proposed so far in the literature for theP|rj; qj|Cmax [1,2,4] to the best of our knowledge, noone has been proven to dominate PLB.The objective of this paper is to introduce the con-

cept of semi-preemptive scheduling and to show how

0167-6377/03/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S0167 -6377(02)00177 -3

50 M. Haouari, A. Gharbi / Operations Research Letters 31 (2003) 49–52

it can be used for deriving a new strong lower boundwhich dominates PLB. However, we <rst start bybrie#y describing the preemptive lower bound whichconstitutes the background of our work.

2. The preemptive lower bound

Given a P|rj; pmtn|Lmax instance, the optimal Lmaxis obtained after repeatedly checking the existence of apreemptive schedule with Lmax equal to an integer trialvalue L. If L− and L+ denote a lower and upper boundon the trial value L, then the optimal Lmax is computedusing a bisection search on the interval [L−; L+].Assume that there exists a preemptive schedule with

Lmax = L. Thus, Cj − dj 6 L for all j ∈ J . That is,the completion time of each job j ∈ J is necessarilyless than or equal to dj + L. This can be assured bysetting a deadline Idj = dj + L for all j ∈ J (i.e. jobj is constrained to be completed before Idj). Horn [5]proposed a polynomial algorithm to test the existenceof a preemptive schedule subject to release dates anddeadlines. This approach consists in transforming thefeasibility problem into an equivalent maximum #owone. For the sake of completeness, we brie#y describeHorn’s procedure.Let {e1; : : : ; eH} be the set containing all release

dates and deadlines of all jobs ranked in increasingorder. A time interval Eh = [eh; eh+1] is de<ned foreach h = 1; : : : ; H − 1. Consider the #ow networkcomposed of job nodes {J1; : : : ; Jn}, interval nodes{E1; : : : ; EH−1}, a source node s, and a sink node t.For each job node Jj (j = 1; : : : ; n), there is an arc(s; Jj) with capacity pj. For each interval node Eh (h=1; : : : ; H − 1), there is an arc (Eh; t) with capacitym(eh+1 − eh). There is an arc (Jj; Eh) with capacityeh+1 − eh if and only if rj 6 eh and eh+1 6 Idj. Apreemptive schedule with Lmax equal to L is de<nedas an assignment of portions of the processing time ofeach job j (j = 1; : : : ; n) to diKerent time intervals Eh(h= 1; : : : ; H − 1). One can prove that such a sched-ule is feasible if and only if the maximum #ow valueis equal to

∑nj=1 pj [5].

The computation of the maximum #ow requiresO(N 3) time, where N is the number of nodes in thenetwork. We have a maximum of 3n+1 nodes. Thus,checking the existence of a preemptive schedule withLmax equal to L requires O(n3) time. By perform-

ing a bisection search on the interval [L−; L+], thelower bound PLB can be computed in O(n3(log n +logpmax)) time [6].A closely related lower bound is based on the

concept of pseudo-preemptive scheduling for theP|rj; qj|Cmax. In a pseudo-preemptive schedule, anyoperation is allowed to be preempted, and each ma-chine can handle several jobs at one time. Moreover,each job can be processed by more than one machineat one time. The Jackson’s pseudo-preemptive sched-ule (JPPS) was introduced by Carlier and Pinson[2]. They show that its makespan, denoted by Cmax(JPPS), is an O(n log n + nm logm) lower bound forthe P|rj; qj|Cmax. The JPPS is a generalization of Jack-son’s preemptive schedule whose makespan is a tightlower bound for 1|rj; qj|Cmax. Its description is ratherlong and may aKect the clarity of the present paper,so we skip it and refer the reader to the original paper.Clearly, �Cmax(JPSS)� −maxj∈J dj is a lower boundfor the P|rj|Lmax. It will be referred hereafter to theJackson’s pseudo-preemptive lower bound (JPPLB).It is worth noting that JPPLB6 PLB. Thus, JPPLB

can be useful for the computation of PLB by settingL− = JPPLB. The upper bound L+ can be computedusing the earliest due date (EDD) rule [1].

3. The semi-preemptive lower bound

In this section, we develop a new lower boundwhich dominates the preemptive bound. The proposedbound is similar in spirit to PLB in the sense that itconsists in checking the feasibility of a schedule withLmax equal to a trial value L.Firstly, by remarking that the latest starting time of

any job j ∈ J is Idj−pj, and its earliest <nishing timeis rj + pj, we can state the following observation:

Observation 1. Assume that there exists a job j suchthat Idj − rj ¡ 2pj. Then, in any non-preemptiveschedule, there is necessarily one machine which mustprocess job j during the interval [ Idj − pj; rj + pj].

According to the above observation, each job j sat-isfying Idj−rj ¡ 2pj is composed of a 6xed and a freepart. Its <xed part is the amount of time 2pj−( Idj−rj)which must be processed in [ Idj −pj; rj +pj], and itsfree part is the amount of time p′

j = Idj − (rj + pj)

M. Haouari, A. Gharbi / Operations Research Letters 31 (2003) 49–52 51

which has to be processed in [rj; Idj−pj]∪[rj+pj; Idj].The other jobs are composed only of a free processingpart p′

j=pj which has to be processed in [rj; Idj]. Thatis, a free part of any job j ∈ J is p′

j = min{pj; Idj −(rj + pj)}.Clearly, a relaxation of the P|rj|Lmax can be ob-

tained if the preemption is allowed for only the freeparts of the jobs in J . We de<ne a semi-preemptiveschedule as a schedule where the <xed parts of thejobs are constrained to start and to <nish at <xed timeswith no preemption, whereas the free parts can be pre-empted.The feasibility of a semi-preemptive schedule with

Lmax equal to a trial value L can be checked as follows:<rst, note that Observation 1 provides a simple wayto compute a lower bound on the number of machineswhich are necessarily loaded at any time. Therefore,the following result clearly holds:

Observation 2. If there is a time t ∈ [0;maxj∈J Idj]such that the number of machines loaded at t isstrictly greater than m, then there is no feasiblesemi-preemptive schedule.

Let S={j ∈ J ; Idj−rj ¡ 2pj} denote the set of jobshaving a <xed processing part. Let e1; e2; : : : ; eK be thediKerent values of rj (j ∈ J ); Idj (j ∈ J ); Idj−pj (j ∈S) and rj + pj (j ∈ S) ranked in increasing order.We denote by mk the number of machines which areidle during the time interval Ek=[ek ; ek+1] (16 k 6K − 1) according to Observation 1. The feasibilityproblem can be solved using the following extensionof Horn’s approach:Consider the #ow network composed of job nodes

{J1; : : : ; Jn}, interval nodes {E1; : : : ; EK−1}, a sourcenode s, and a sink node t. For each job node Jj (j =1; : : : ; n) such that p′

j ¿ 0, there is an arc (s; Jj) withcapacityp′

j representing the free part of job j. For eachk = 1; : : : ; K − 1, there is an arc (Ek; t) with capacitymk(ek+1 − ek). There is an arc (Jj; Ek) with capacityek+1 − ek if and only if one of the three followingconditions is satis<ed:

• Idj − rj ¡ 2pj; rj 6 ek and ek+1 6 Idj − pj,• Idj − rj ¡ 2pj; rj + pj 6 ek and ek+1 6 Idj,• Idj − rj ¿ 2pj; rj 6 ek and ek+1 6 Idj.

Obviously, an interval node Ek withmk=0 or whichis not connected with any job node is dropped from

Table 1Data of a three job-2 machine instance

j 1 2 3

rj 23 0 0pj 14 45 74dj 59 68 111

Fig. 1. An optimal preemptive schedule with Lmax =−22.

the network. Clearly, a semi-preemptive schedule withLmax equal to L exists if and only if the maximum #owis equal to

∑j∈J p

′j.

We have a maximum of 5n + 1 nodes in the cor-responding network. Therefore, the feasibility testrequires O(n3) time. The semi-preemptive lowerbound, denoted by SPLB, is de<ned as the optimalsemi-preemptive Lmax obtained after performing abisection search on the interval [L−; L+]. The boundSPLB can be computed in O(n3(log n+ logpmax)).Clearly, we have

PLB6 SPLB:

Interestingly, for some instances SPLB strictly domi-nates PLB. This can be demonstrated with the follow-ing example.

Example. Consider the three job-2 machine instancede<ned by Table 1.

We have L−=JPPLB=−22 and L+=0. An optimalpreemptive schedule with Lmax = −22 is depicted inFig. 1. Therefore, PLB =−22.Assume that there exists a semi-preemptive sched-

ule with Lmax equal to the trial value L = −22.Thus, the deadlines are set to Id1 = 37, Id2 =46 and Id3 = 89. According to Observation 1,there is necessarily one machine which must process

52 M. Haouari, A. Gharbi / Operations Research Letters 31 (2003) 49–52

Fig. 2. An optimal semi-preemptive schedule with Lmax =−7.

job 1 in the interval [23; 37] for 14 units of time. Also,a second machine must process job 2 in the interval[1; 45] for 44 units of time, and a third machine mustprocess job 3 in the interval [15; 74] for 59 units oftime. That is, there must be three loaded machinesin the interval [23; 37]. Therefore, the instance isinfeasible (cf. Observation 2).Consider the trial value L=−8. The deadlines are

set to Id1 = 51, Id2 = 60 and Id3 = 103. According toObservation 1, there is necessarily one machine whichmust process job 2 in the interval [15; 45] for 30 unitsof time, and a second machine which must processjob 3 in the interval [29; 74] for 45 units of time.Thus, the free processing parts are p′

1 = 14, p′2 = 15

and p′3 = 29. The maximum #ow value is equal to

56 whereas∑

j∈J p′j = 58. Therefore, the instance is

infeasible.

Now, assume that L = −7. The deadlines areset to Id1 =52, Id2 = 61 and Id3 = 104. According toObservation 1, there is necessarily one machine whichmust process job 2 in the interval [16; 45] for 29 unitsof time, and a second machine which must processjob 3 in the interval [30; 74] for 44 units of time.Thus, the free processing parts are p′

1 = 14; p′2 = 16

and p′3 = 30. The maximum #ow value is equal to∑

j∈J p′j = 60. Therefore, SPLB = −7. An optimal

semi-preemptive schedule with Lmax =−7 is depictedin Fig. 2.

References

[1] J. Carlier, Scheduling jobs with release dates and tails onidentical machines to minimize the makespan, EuropeanJ. Oper. Res. 29 (1987) 298–306.

[2] J. Carlier, E. Pinson, Jackson’s pseudo preemptive schedulefor the Pm|rj ; qj|Cmax scheduling problem, Ann. Oper. Res.83 (1998) 41–58.

[3] M.R. Garey, D.S. Johnson, Computers and Intractability:A Guide to the Theory of NP-Completeness, Freeman, SanFransisco, 1979.

[4] A. Gharbi, M. Haouari, Minimizing makespan on parallelmachines subject to release dates and delivery times,J. Scheduling, accepted for publication.

[5] W.A. Horn, Some simple scheduling algorithms, Naval Res.Logistics Quart. 21 (1974) 177–185.

[6] J. Labetoulle, E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan,Preemptive scheduling of uniform machines subject to releasedates, Progress in Combinatorial Optimization, AcademicPress, Florida, 1984, pp. 245–261.