Abstract
In the past two decades, scheduling with machine availability constraints has received more and more attention. Until now most research has focused on the setting where all machine unavailability information is known at the beginning of scheduling horizon. In real world, this is impractical in some cases.
In this article, we consider the situation where the scheduler has to make scheduling decisions without any knowledge of the machine unavailable intervals. In particular, we study the problem of minimizing the total weighted completion time. When there are two or more unavailable intervals on a single machine, Fu et al. (2009) have shown that the problem is exponentially inapproximable even when jobs’ weights are equal to their processing times and one has full knowledge of unavailability. So in this paper we consider the scheduling problem on a single machine with a single unavailable period. And we assume that every job has a weight proportional to its processing time. Based on whether the unavailable interval is due to a breakdown or an emergent job, we have breakdown model and emergent job model. We first show that no \(\tfrac{\sqrt{5}+1}{2}\)-competitive online algorithm exists for breakdown model, and no \(\tfrac{11-\sqrt{2}}{7}\)-competitive online algorithm exists for emergent job model. Then we show that the simple LPT (Largest Processing Time first) rule can give a 2-competitive ratio and 9/5-competitive ratio for breakdown model and emergent job model, respectively. We show the ratios are tight by examples. For offline case, we show that First Fit LPT (FF-LPT) rule can give a tight approximation ratio of 2 and 4/3 for breakdown model and emergent job model, respectively. Finally, our experimental results show that in practice, both LPT and FF- LPT perform very well and the performance improves when the number of jobs n increases. When n ≥ 50, the worst error ratio of LPT is about 8.7 %, and the worst error ratio of FF-LPT is about 0.7%. So in both cases, the error ratio is quite far from the theoretical bound.
This work is supported by PSC-CUNY Research Award.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Adiri, I., Bruno, J., Frostig, E., Rinnooy Kan, A.H.G.: Single machine flowtime scheduling with a single breakdown. Acta Informatica 26, 679–696 (1989)
Albers, S., Schmidt, G.: Scheduling with Unexpected Machine Breakdowns. In: Hochbaum, D.S., Jansen, K., Rolim, J.D.P., Sinclair, A. (eds.) RANDOM 1999 and APPROX 1999. LNCS, vol. 1671, pp. 269–280. Springer, Heidelberg (1999)
Arkin, R., Roundy, R.: Weighted tardiness scheduling on parallel machines with proportional weights. Operations Research 39, 64–81 (1991)
Breit, J.: Improved approximation for non-preemptive single machine flowtime scheduling with an availability constraint. European Journal of Operational Research 183(3), 516–524 (2007)
Diedrich, F., Schwarz, U.M.: A Framework for Scheduling with Online Availability. In: Kermarrec, A.-M., Bougé, L., Priol, T. (eds.) Euro-Par 2007. LNCS, vol. 4641, pp. 205–213. Springer, Heidelberg (2007)
Diedrich, F., Jansen, K.: Improved approximation algorithms for scheduling with fixed jobs. In: Proceeding of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 675–684 (2009)
Fu, B., Huo, Y., Zhao, H.: Exponential inapproximability and FPTAS for scheduling with availability constraints. Theoretical Computer Science 410, 2663–2674 (2009)
Fu, B., Huo, Y., Zhao, H.: Approximation schemes for parallel machine scheduling with availability constraints. Discrete Applied Mathematics 159(15), 1555–1565 (2011)
He, Y., Zhong, W., Gu, H.: Improved algorithms for two single machine scheduling problems. Theoretical Computer Science 363, 257–265 (2006)
Kacem, I.: Approximation algorithm for the weighted flow-time minimization on a single machine with a fixed non-availability interval. Computers & Industrial Engineering 54, 401–410 (2008)
Kacem, I., Chu, C.: Worst-case analysis of the WSPT and MWSPT rules for single machine scheduling with one planned setup period. European Journal of Operational Research 187(3), 1080–1089 (2008)
Kacem, I., Mahjoub, R.: Fully polynomial time approximation scheme for the weighted flow-time minimization on a single machine with a fixed non-availability interval. Computers & Industrial Engineering 56(4), 1708–1712 (2009)
Kellerer, H., Strusevich, V.A.: Fully polynomial approximation schemes for a symmetric quadratic knapsack problem and its scheduling applications. Algorithmica 57(4), 769–795 (2010)
Lee, C.Y.: Machine scheduling with an availability constraints. Journal of Global Optimization 9, 363–382 (1996)
Lee, C.Y., Liman, S.D.: Single machine flow-time scheduling with scheduled maintenance. Acta Informatica 29(4), 375–382 (1992)
Ma, Y., Chu, C., Zuo, C.: A survey of scheduling with deterministic machine availability constraints. Computers & Industrial Engineering 58(2), 199–211 (2010)
Sadfi, C., Penz, B., Rapine, C., Blazewicz, J., Formanowicz, P.: An improved approximation algorithm for the single machine total completion time scheduling problem with availability constraints. European Journal of Operational Research 161, 3–10 (2005)
Scharbrodt, M., Steger, A., Weisser, H.: Approximability of scheduling with fixed jobs. Journal of Scheduling 2, 267–284 (1999)
Tan, Z., He, Y.: Optimal online algorithm for scheduling on two identical machines with machine availability constraints. Information Processing Letters 83, 323–329 (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Huo, Y., Reznichenko, B., Zhao, H. (2012). Minimizing Total Weighted Completion Time with Unexpected Machine Unavailability. In: Lin, G. (eds) Combinatorial Optimization and Applications. COCOA 2012. Lecture Notes in Computer Science, vol 7402. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31770-5_26
Download citation
DOI: https://doi.org/10.1007/978-3-642-31770-5_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31769-9
Online ISBN: 978-3-642-31770-5
eBook Packages: Computer ScienceComputer Science (R0)