Abstract
Motivated by mail delivery scheduling problems arising in Royal Mail, we study a generalization of the fundamental makespan scheduling problem \(P||C_{\max }\) which we call the Bounded Job Start Scheduling Problem. Given a set of jobs, each one specified by an integer processing time \(p_j\), 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 \(g\le m\) on the number of jobs that may simultaneously begin per unit of time. We show that Longest Processing Time First (LPT) algorithm is tightly 2-approximate. After proving that the problem is strongly \(\mathcal {NP}\)-hard even when \(g=1\), we elaborate on improving the 2-approximation ratio for this case. We distinguish the classes of long and short instances satisfying \(p_j\ge m\) and \(p_j<m\), 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 solutions with tightly satisfied packing and bounded job start constraints. For a broad family of instances excluding degenerate instances with many very long jobs and instances with few machines, we derive a 1.985-approximation ratio. For general instances, we require machine augmentation to obtain better than 2-approximate schedules. Finally, we exploit machine augmentation and a state-of-the-art lexicographic optimization method for \(P||C_{\max }\) under uncertainty to propose a two-stage robust optimization approach for bounded job start scheduling under uncertainty attaining good trade-offs in terms of makespan and number of used machines. We substantiate this approach numerically using Royal Mail data.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bertsimas, D., Sim, M.: The price of robustness. Oper. Res. 52(1), 35–53 (2004)
Billaut, J.-C., Sourd, F.: Single machine scheduling with forbidden start times. 4OR 7(1), 37–50 (2009)
Gabay, M., Rapine, C., Brauner, N.: High-multiplicity scheduling on one machine with forbidden start and completion times. J. Sched. 19(5), 609–616 (2016)
Goerigk, M., Schöbel, A.: Algorithm engineering in robust optimization. In: Kliemann, L., Sanders, P. (eds.) Algorithm Engineering. LNCS, vol. 9220, pp. 245–279. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-49487-6_8
Graham, R.L.: Bounds on multiprocessing timing anomalies. SIAM J. Appl. Math. 17(2), 416–429 (1969)
Letsios, D., Ruth, M.: Exact lexicographic scheduling and approximate rescheduling. arXiv 1805.03437 (2018)
Liebchen, C., Lübbecke, M., Möhring, R., Stiller, S.: The concept of recoverable robustness, linear programming recovery, and railway applications. In: Ahuja, R.K., Möhring, R.H., Zaroliagis, C.D. (eds.) Robust and Online Large-Scale Optimization. LNCS, vol. 5868, pp. 1–27. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-05465-5_1
Mnich, M., Bevern, R.V.: Parameterized complexity of machine scheduling: 15 open problems. Comput. Oper. Res. 100, 254–261 (2018)
Rapine, C., Brauner, N.: A polynomial time algorithm for makespan minimization on one machine with forbidden start and completion times. Discrete Optim. 10(4), 241–250 (2013)
Schäffter, M.W.: Scheduling with forbidden sets. Discrete Appl. Math. 72(1–2), 155–166 (1997)
Skutella, M., Verschae, J.: Robust polynomial-time approximation schemes for parallel machine scheduling with job arrivals and departures. Math. Oper. Res. 41(3), 991–1021 (2016)
Svensson, O.: Hardness of precedence constrained scheduling on identical machines. SIAM J. Comput. 40(5), 1258–1274 (2011)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Bradley, J.T., Letsios, D., Misener, R., Page, N. (2019). Approximating Bounded Job Start Scheduling with Application in Royal Mail Deliveries Under Uncertainty. In: Li, Y., Cardei, M., Huang, Y. (eds) Combinatorial Optimization and Applications. COCOA 2019. Lecture Notes in Computer Science(), vol 11949. Springer, Cham. https://doi.org/10.1007/978-3-030-36412-0_6
Download citation
DOI: https://doi.org/10.1007/978-3-030-36412-0_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-36411-3
Online ISBN: 978-3-030-36412-0
eBook Packages: Computer ScienceComputer Science (R0)