On-line scheduling on a batch machine to minimize makespan with limited restarts

Fu, Ruyan; Tian, Jing; Yuan, Jinjiang; He, Cheng

doi:10.1016/j.orl.2007.07.001

Cited by 18 publications

(8 citation statements)

References 7 publications

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“…Limited restart is first introduced by Fu et al [6]. Limited restart means that once a batch contains some restarted jobs, we cannot restart the batch again.…”

Section: Mathematical Problems In Engineeringmentioning

confidence: 99%

Online Scheduling with Delivery Time on a Bounded Parallel Batch Machine with Limited Restart

Liu

Wan

Yan

et al. 2015

Mathematical Problems in Engineering

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We consider the online (over time) scheduling of equal length jobs on a bounded parallel batch machine with batch capacitybto minimize the time by which all jobs have been delivered with limited restart. Here, “restart” means that a running batch may be interrupted, losing all the work done on it, and jobs in the interrupted batch are then released and become independently unscheduled jobs, called restarted jobs. “Limited restart” means that a running batch which contains some restarted jobs cannot be restarted again. Whenb=2, we propose a best possible online algorithmH(b=2)with a competitive ratio of1+α, whereαis the positive solution of2α(1+α)=1. Whenb≥3, we present a best possible online algorithmH(b≥3)with a competitive ratio of1+β, whereβis the positive solution ofβ(1+β)2=1.

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“…Limited restart is first introduced by Fu et al [6]. Limited restart means that once a batch contains some restarted jobs, we cannot restart the batch again.…”

Section: Mathematical Problems In Engineeringmentioning

confidence: 99%

Online Scheduling with Delivery Time on a Bounded Parallel Batch Machine with Limited Restart

Liu

Wan

Yan

et al. 2015

Mathematical Problems in Engineering

Self Cite

View full text Add to dashboard Cite

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“…For the scenario where restart is allowed, Fu et al (2007) showed that for the unbounded model to minimize makespan, there is no online algorithm with competitive ratio less than 5− √ 5 2 , and they also provided an upper bound of 3 2 . Fu et al (2008) studied a special case with limited restarts such that any batch containing a job which has ever been restarted cannot be preempted, presenting an optimal 3 2 -competitive algorithm. considered family job constraint in the single batch machine scheduling problem.…”

Section: Introductionmentioning

confidence: 99%

Optimal Semi-Online Algorithm for Scheduling on Two Parallel Batch Processing Machines

Liu

Zheng

Zhu

et al. 2014

Asia Pac. J. Oper. Res.

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1450038-1 Asia Pac. J. Oper. Res. 2014.31. Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 02/03/15. For personal use only. M. Liu et al.to minimize the makespan. Given any job J j together with its following job J j+1 , it is assumed that their processing times satisfy p j+1 ≥ αp j where α ≥ 1 is a constant. That is, jobs arrive in a non-decreasing order of processing times. We mainly propose an optimal φ-competitive online algorithm where φ ≥ 1 is a solution of equation Proof. ByStep 2 of the algorithm, we have the desired result.Proof. By Corollary 1, r n > S n−1 and then C max (π) ≥ r n + p n > S n−1 + p n . Since J n is regular, we have C max (σ) = S n + p n = max{φ(S n−1 + p n ), r n + p n }, implying C max (σ)/C max (π) ≤ φ. The lemma follows.By the above lemma, we focus on proving C max (σ)/C max (π) ≤ φ in the other case where J n is a delayed job in the remainder. By symmetry of two machines, we assume without loss of generality that J n is scheduled on M 1 . Lemma 4. n ≥ 3. 1450038-7 Asia Pac. J. Oper. Res. 2014.31. Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 02/03/15. For personal use only. M. Liu et al.Proof. We already have n ≥ 2. Assume otherwise that n = 2. On the arrival of the second job J 2 , there is only one of the two machines processing the first job J 1 , implying that the other machine is being idle from the beginning and thus J 2 cannot be postponed by job J 1 . A contradiction to the assumption. The lemma follows. Theorem 2. AlgorithmProof. We prove the theorem by contradiction. Assume otherwise C max (σ)/ C max (π) > φ. Consider the last three jobs J n−2 , J n−1 and J n in σ with C n−2 < C n−1 < C n due to Corollary 2. There are two cases below.Case 1. J n−1 is scheduled on M 1 . We further discuss two subcases according to the scheduling of job J n−2 .Case 1.1. J n−2 is scheduled on M 1 . By Corollary 1, r n > S n−1 , which is not smaller than C n−2 due to case condition that both J n−2 and J n−1 are scheduled on M 1 with S n−2 < S n−1 . Together with Corollary 2, we claim that r n > C n−2 > C i for 1 ≤ i < n − 2, and thus machine M 2 is idle and available from time r n , implying J n is a regular job. By Lemma 3, C max (σ)/C max (π) ≤ φ. A contradiction.Case 1.2. J n−2 is scheduled on M 2 . By Corollary 2, we have C n−2 < C n−1 and thus J n must be scheduled on M 2 by Step 2.1 of the algorithm. A contradiction to the fact that J n is scheduled on machine M 1 .Case 2. J n−1 is scheduled on M 2 . We similarly divide this case into two subcases.Case 2.1. J n−2 is scheduled on M 2 . Let J l (1 ≤ l < n − 2) be the preceding job of J n on machine M 1 . We claim such job J l does exist since otherwise it contradicts the fact that J n is a delayed job. With similar reasoning to Case 1.1, we have r n > S n−1 ≥ C n−2 > C l , implying that J n is a regular job. A contradiction again.Case 2.2. J n−2 is scheduled on M 1 . Since J n−2 and J n are both scheduled on M 1 , combining C n−2 < C n−1 and the fact that J n is a delayed job, we have S n = C...

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“…In this paper a running batch is allowed to have limited restarts [4]. This means that once a batch contains some restarted jobs, we cannot restart the batch again.…”

Section: Introductionmentioning

confidence: 99%

“…They provided a best possible online algorithm with a competitive ratio 5− √ 5 2 . Fu et al [4] studied the same problem with limited restarts. They proved that the best online algorithm is 3 2 -competitive.…”

Section: Introductionmentioning

confidence: 99%

On-line scheduling on a batch machine to minimize makespan with limited restarts

Cited by 18 publications

References 7 publications

Online Scheduling with Delivery Time on a Bounded Parallel Batch Machine with Limited Restart

Online Scheduling with Delivery Time on a Bounded Parallel Batch Machine with Limited Restart

Optimal Semi-Online Algorithm for Scheduling on Two Parallel Batch Processing Machines

Online scheduling on two parallel-batching machines with limited restarts to minimize the makespan

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