High-multiplicity scheduling on one machine with forbidden start and completion times

Gabay, Michaël; Rapine, Christophe; Brauner, Nadia

doi:10.1007/s10951-016-0475-z

Cited by 4 publications

(3 citation statements)

References 12 publications

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“…Even when there is a strict order between jobs in the same forbidden set, the scheduling with forbidden sets problem is equivalent to the precedence-constrained scheduling problem P| pr ec|C max and cannot be approximated by a factor lower than (2 − ), assuming a variant of the unique games conjecture (Svensson 2011). Also, BJSP relaxes the scheduling with forbidden job start times problem, where no job may begin at certain time points, which does not admit constantfactor approximation algorithms (Billaut and Sourd 2009;Gabay et al 2016; Mnich and van Bevern 2018; Rapine and Brauner 2013). Despite the commonalities with the aforementioned literature, to the authors' knowledge, there is a lack of approximation algorithms for scheduling problems with bounded job starts.…”

Section: Related Workmentioning

confidence: 99%

Approximate and robust bounded job start scheduling for Royal Mail delivery offices

Letsios

Bradley²,

Suraj

et al. 2021

J Sched

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Motivated by mail delivery scheduling problems arising in Royal Mail, we study a generalization of the fundamental makespan scheduling $$P||C_{\max }$$ P | | C max problem which we call the bounded job start scheduling problem. Given a set of jobs, each specified by an integer processing time $$p_j$$ p j , that have to be executed non-preemptively by a set of m parallel identical machines, the objective is to compute a minimum makespan schedule subject to an upper bound $$g\le m$$ g ≤ m on the number of jobs that may simultaneously begin per unit of time. With perfect input knowledge, we show that Longest Processing Time First (LPT) algorithm is tightly 2-approximate. After proving that the problem is strongly $${\mathcal {N}}{\mathcal {P}}$$ N P -hard even when $$g=1$$ g = 1 , we elaborate on improving the 2-approximation ratio for this case. We distinguish the classes of long and short instances satisfying $$p_j\ge m$$ p j ≥ m and $$p_j<m$$ p j < m , respectively, for each job j. We show that LPT is 5/3-approximate for the former and optimal for the latter. Then, we explore the idea of scheduling long jobs in parallel with short jobs to obtain tightly satisfied packing and bounded job start constraints. For a broad family of instances excluding degenerate instances with many very long jobs, we derive a 1.985-approximation ratio. For general instances, we require machine augmentation to obtain better than 2-approximate schedules. In the presence of uncertain job processing times, we exploit machine augmentation and lexicographic optimization, which is useful for $$P||C_{\max }$$ P | | C max under uncertainty, to propose a two-stage robust optimization approach for bounded job start scheduling under uncertainty aiming in a low number of used machines. Given a collection of schedules of makespan $$\le D$$ ≤ D , this approach allows distinguishing which are the more robust. We substantiate both the heuristics and our recovery approach numerically using Royal Mail data. We show that for the Royal Mail application, machine augmentation, i.e., short-term van rental, is especially relevant.

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Section: Related Workmentioning

confidence: 99%

Approximate and robust bounded job start scheduling for Royal Mail delivery offices

Letsios

Bradley²,

Suraj

et al. 2021

J Sched

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“…For makespan minimization on a single machine, Billaut and Sourd (2009) gave an algorithm that runs in n O(τ 2 ) time for τ forbidden start times and n jobs; this was improved by Rapine and Brauner (2013) to n O(τ) time. For the high-multiplicity encoding of the input-given by binary numbers n t encoding the number of jobs having the same forbidden start and end times- Gabay et al (2016) showed a polynomial-time algorithm if the number τ of forbidden times is constant. All of these results leave open the possibility for fixed-parameter tractability of the problem parameterized by τ.…”

Section: Forbidden Start and End Timesmentioning

confidence: 99%

“…Gabay et al (2016) showed a polynomial-time algorithm if the number τ of forbidden times is constant. All of these results leave open the possibility for fixed-parameter tractability of the problem parameterized by τ.…”

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confidence: 99%

Parameterized complexity of machine scheduling: 15 open problems

Mnich

Bevern

2018

Computers & Operations Research

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Machine scheduling problems are a long-time key domain of algorithms and complexity research. A novel approach to machine scheduling problems are fixed-parameter algorithms. To stimulate this thriving research direction, we propose 15 open questions in this area whose resolution we expect to lead to the discovery of new approaches and techniques both in scheduling and parameterized complexity theory.

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Approximating Bounded Job Start Scheduling with Application in Royal Mail Deliveries Under Uncertainty

Bradley¹,

Letsios

Misener

et al. 2019

Combinatorial Optimization and Applications

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High-multiplicity scheduling on one machine with forbidden start and completion times

Cited by 4 publications

References 12 publications

Approximate and robust bounded job start scheduling for Royal Mail delivery offices

Approximate and robust bounded job start scheduling for Royal Mail delivery offices

Parameterized complexity of machine scheduling: 15 open problems

Approximating Bounded Job Start Scheduling with Application in Royal Mail Deliveries Under Uncertainty

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