Journal of Combinatorial Optimization

, Volume 32, Issue 3, pp 885–905 | Cite as

Scheduling arc shut downs in a network to maximize flow over time with a bounded number of jobs per time period

  • Natashia Boland
  • Thomas Kalinowski
  • Simranjit Kaur


We study the problem of scheduling maintenance on arcs of a capacitated network so as to maximize the total flow from a source node to a sink node over a set of time periods. Maintenance on an arc shuts down the arc for the duration of the period in which its maintenance is scheduled, making its capacity zero for that period. A set of arcs is designated to have maintenance during the planning period, which will require each to be shut down for exactly one time period. In general this problem is known to be NP-hard, and several special instance classes have been studied. Here we propose an additional constraint which limits the number of maintenance jobs per time period, and we study the impact of this on the complexity.


Network models Complexity theory Maintenance scheduling Mixed integer programming 

Mathematics Subject Classification

90C10 90B10 68Q25 



We would like to thank two anonymous referees for valuable comments that significantly improved the presentation of our results, in particular the proof of Proposition 2. This research was supported by the ARC Linkage Grants Nos. LP0990739 and LP1102000524 and HVCCC P/L.


  1. Baxter M, Elgindy T, Ernst AT, Kalinowski T, Savelsbergh MWP (2014) Incremental network design with shortest paths. Eur J Oper Res 238(3):675–684MathSciNetCrossRefzbMATHGoogle Scholar
  2. Boland N, Kalinowski T, Kapoor R, Kaur S (2014) Scheduling unit time arc shutdowns to maximize network flow over time: complexity results. Networks 63(2):196–202MathSciNetCrossRefGoogle Scholar
  3. Boland N, Kalinowski T, Waterer H, Zheng L (2011) An optimisation approach to maintenance scheduling for capacity alignment in the Hunter Valley coal chain. In: Baafi EY, Kininmonth RJ, Porter I (eds) Proceedings of the 35th APCOM symposium: applications of computers and operations research in the minerals industry, The Australasian Institute of Mining and Metallurgy Publication Series, pp 887–897Google Scholar
  4. Boland N, Kalinowski T, Waterer H, Zheng L (2013) Mixed integer programming based maintenance scheduling for the Hunter Valley coal chain. J Sched 16(6):649–659MathSciNetCrossRefGoogle Scholar
  5. Boland N, Kalinowski T, Waterer H, Zheng L (2014) Scheduling arc maintenance jobs in a network to maximize total flow over time. Discret Appl Math 163(1):34–52MathSciNetCrossRefzbMATHGoogle Scholar
  6. Boland N, McGowan B, Mendes A, Rigterink F (2013) Modelling the capacity of the Hunter Valley coal chain to support capacity alignment of maintenance activities. In: Piantadosi J, Anderssen RS, Boland J (eds) MODSIM2013, 20th international congress on modelling and simulation, Modelling and Simulation Society of Australia and New Zealand, pp 3302–3308Google Scholar
  7. Boland N, Savelsbergh MWP (2011) Optimizing the Hunter Valley coal chain. In: Gurnani H, Mehrotra A, Ray S (eds) Supply chain disruptions: theory and practice of managing risk. Springer-Verlag, London Ltd., LondonGoogle Scholar
  8. Edmonds J (1965) Paths, trees, and flowers. Can J Math 17(3):449–467MathSciNetCrossRefzbMATHGoogle Scholar
  9. Gabow HN (1990) Data structures for weighted matching and nearest common ancestors with linking. In: Proceedings of the 1st ACM-SIAM symposium on discrete algorithms, SODA 1990, pp 434–443Google Scholar
  10. Kalinowski T, Matsypura D, Savelsbergh MWP (2015) Incremental network design with maximum flows. Eur J Oper Res 242(1):51–62MathSciNetCrossRefzbMATHGoogle Scholar
  11. King V, Rao S, Tarjan R (1994) A faster deterministic maximum flow algorithm. J Algorithms 17(3):447–474MathSciNetCrossRefzbMATHGoogle Scholar
  12. Koch R, Nasrabadi E, Skutella M (2011) Continuous and discrete flows over time. Math Methods Oper Res 73:301–337MathSciNetCrossRefzbMATHGoogle Scholar
  13. Kotnyek B (2003) An annotated overview of dynamic network flows. Technical Report 4936, INRIAGoogle Scholar
  14. Lidén T (2014) Survey of railway maintenance activities from a planning perspective and literature review concerning the use of mathematical algorithms for solving such planning and scheduling problems. Technical report, Linköpings universitetGoogle Scholar
  15. Nurre SG (2013) Integrated network design and scheduling problems: Optimization algorithms and applications. PhD thesis, Rensselaer Polytechnic Institute. online:
  16. Nurre SG, Cavdaroglu B, Mitchell JE, Sharkey TC, Wallace WA (2012) Restoring infrastructure systems: an integrated network design and scheduling (INDS) problem. Eur J Oper Res 223(3):794–806MathSciNetCrossRefzbMATHGoogle Scholar
  17. Nurre SG, Sharkey TC (2014) Integrated network design and scheduling problems with parallel identical machines: complexity results and dispatching rules. Networks 63:306–326MathSciNetCrossRefGoogle Scholar
  18. Orlin JB (2013) Max flows in \(O(nm)\) time, or better. In: Proceedings of the 45th ACM symposium on theory of computing (STOC 2013), ACM, pp 765–774Google Scholar
  19. Pinedo M (2008) Scheduling: theory, algorithms, and systems. Springer, New YorkzbMATHGoogle Scholar
  20. Skutella M (2009) An introduction to network flows over time. In: Cook W, Lovasz L, Vygen J (eds) Research trends in combinatorial optimization. Springer, Berlin, pp 451–482CrossRefGoogle Scholar
  21. Tawarmalani M, Li Y (2011) Multi-period maintenance scheduling of tree networks with minimum flow disruption. Nav Res Logist 58(5):507–530MathSciNetCrossRefzbMATHGoogle Scholar
  22. Williamson DP, Shmoys DB (2011) The design of approximation algorithms. Cambridge University Press, New YorkCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Natashia Boland
    • 1
    • 2
  • Thomas Kalinowski
    • 1
  • Simranjit Kaur
    • 1
  1. 1.University of NewcastleNewcastleAustralia
  2. 2.H. Milton Stewart School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA

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