An O(n2) algorithm for scheduling equal-length preemptive jobs on a single machine to minimize total tardiness

Tian, Zhongjun; Ng, C. T.; Cheng, T.C.E.

doi:10.1007/s10951-006-7039-6

Cited by 22 publications

(12 citation statements)

References 7 publications

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“…Problem 1 | r j , p j = p, pmtn | T j can be solved in O(n 2 ) time (Tian et al 2006). In Du and Leung (1990), it has been shown that problems 1 | pmtn | T j and 1 T j are NP-hard in the ordinary sense whereas in Lawler (1977), a pseudopolynomial algorithm has been proposed for these problems.…”

Section: Introductionmentioning

confidence: 99%

Minimizing total tardiness on parallel machines with preemptions

Kravchenko

Werner

2010

J Sched

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The basic scheduling problem we are dealing with in this paper is the following one. A set of jobs has to be scheduled on a set of parallel uniform machines. Each machine can handle at most one job at a time. Each job becomes available for processing at its release date. All jobs have the same execution requirement and arbitrary due dates. Each machine has a known speed. The processing of any job may be interrupted arbitrarily often and resumed later on any machine. The goal is to find a schedule that minimizes the sum of tardiness, i.e., we consider problem Q | r j , p j = p, pmtn | T j whose complexity status was open. Recently, Tian et al. (J. Sched. 9:343-364, 2006) proposed a polynomial algorithm for problem 1 | r j , p j = p, pmtn | T j . We show that both the problem P | pmtn | T j of minimizing total tardiness on a set of parallel machines with allowed preemptions and the problem P | r j , p j = p, pmtn | T j of minimizing total tardiness on a set of parallel machines with release dates, equal processing times and allowed preemptions are NP-hard. Moreover, we give a polynomial algorithm for the case of uniform machines without release dates, i.e., for problem Q | p j = p, pmtn | T j .

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Section: Introductionmentioning

confidence: 99%

Minimizing total tardiness on parallel machines with preemptions

Kravchenko

Werner

2010

J Sched

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“…Inequalities (77) and (78) are surely true, but they do not really matter here. It appears that both values (7,5) Apart from the artifact, some regularities are seen in both Figs. 2 and 3.…”

Section: Computational Studymentioning

confidence: 82%

“…Models based on the branch-and-bound approach are commonly used for that [5,6]. For tight-tardy progressive single machine scheduling, where release dates are set at nonrepeating integers from 1 through the total number of jobs, and due dates are tightly set after the respective release dates (although sometimes a few jobs can be completed without tardiness), the exact model simplifies owing to no weights are included and each job has the same number of processing periods [7]. An open question is whether the exact schedule computation time changes if the release dates are input to the model in reverse order.…”

Section: Introductionmentioning

confidence: 99%

Efficient Exact Minimization of Total Tardiness in Tight-Tardy Progressive Single Machine Scheduling With Idling-Free Preemptions of Equal-Length Jobs

Romanuke

2020

KPI SN

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Background. A schedule ensuring the exactly minimal total tardiness can be found with the respective integer linear programming problem. An open question is whether the exact schedule computation time changes if the job release dates are input to the model in reverse order. Objective. The goal is to ascertain whether the job order in tight-tardy progressive single machine scheduling with idling-free preemptions of equal-length jobs influences the speed of computing the exact solution. The Boolean linear programming model provided for finding schedules with the minimal total tardiness is used. Methods. To achieve the said goal, a computational study is carried out with a purpose to estimate the averaged computation time for both ascending and descending orders of job release dates. Instances of the job scheduling problem are generated so, that schedules which can be obtained trivially, without the exact model, are excluded. Results. Based on the three-dimensional barred plot of the relative difference between the averaged computation times, it has been shown that a possibility exists to find schedules more efficiently by manipulating the job order. For instance, schedules of 5 jobs consisting of two processing periods each are found on average by 14.67 % faster for the descending job order. In another example of 7 three-parted jobs, an optimal schedule is found on average in 69.51 seconds by the ascending job order, whereas the descending job order takes just 36.52 seconds to find it, saving thus 32.99 seconds. Conclusions. Scheduling a fewer jobs divided into a fewer job parts is executed on average faster by the descending job order. As the number of jobs increases along with increasing the number of their processing periods, the ascending job order becomes more efficient. However, the computation time efficiency by both job orders tends to be irregular.

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“…For the criteria related to tardiness T j and unit penalty U j , it is surprising to see that preemptive problems with precedence constraints have not been studied yet. More precisely, Tian et al [2006] have shown that 1|pmtn, r j , p j = p| T j is solvable in polynomial time, but it is still open whether adding precedence constraints leads this problem to be N P-hard or not; the problem remains also open when considering the more general criterion w j T j . The same outline appears when considering unit penalty: 1|r j ; pmtn; p j = p| w j U j is solvable in polynomial time (with an algorithm in O(n 10 ) by Baptiste [1999], and in O(n 4 ) by Baptiste et al [2004b]); nevertheless, nothing has been shown when adding precedence constraints to the problem.…”

Section: Minimum Np-hard Casesmentioning

confidence: 99%

A survey on how the structure of precedence constraints may change the complexity class of scheduling problems

Prot¹,

Bellenguez‐Morineau²

2017

J Sched

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This survey aims at demonstrating that the structure of precedence constraints plays a tremendous role on the complexity of scheduling problems. Indeed many problems can be N P-hard when considering general precedence constraints, while they become polynomially solvable for particular precedence constraints. Add to this, the existence of many very exciting challenges in this research area is underlined.

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An O(n2) algorithm for scheduling equal-length preemptive jobs on a single machine to minimize total tardiness

Cited by 22 publications

References 7 publications

Minimizing total tardiness on parallel machines with preemptions

Minimizing total tardiness on parallel machines with preemptions

Efficient Exact Minimization of Total Tardiness in Tight-Tardy Progressive Single Machine Scheduling With Idling-Free Preemptions of Equal-Length Jobs

A survey on how the structure of precedence constraints may change the complexity class of scheduling problems

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