Skip to main content
Log in

On solving continuous-time dynamic network flows

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

Temporal dynamics is a crucial feature of network flow problems occurring in many practical applications. Important characteristics of real-world networks such as arc capacities, transit times, transit and storage costs, demands and supplies etc. are subject to fluctuations over time. Consequently, also flow on arcs can change over time which leads to so-called dynamic network flows. While time is a continuous entity by nature, discrete-time models are often used for modeling dynamic network flows as the resulting problems are in general much easier to handle computationally. In this paper, we study a general class of dynamic network flow problems in the continuous-time model, where the input functions are assumed to be piecewise linear or piecewise constant. We give two discrete approximations of the problem by dividing the considered time range into intervals where all parameters are constant or linear. We then present two algorithms that compute, or at least converge to optimum solutions. Finally, we give an empirical analysis of the performance of both algorithms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ahuja R.K., Magnanti T.L., Orlin J.B.: Network Flows. Theory, Algorithms, and Applications. Prentice Hall, Englewood Cliffs (1993)

    Google Scholar 

  2. Anderson, E.J.: A Continuous Model for Job-Shop Scheduling. PhD thesis, University of Cambridge (1978)

  3. Anderson E.J.: Extreme-points for continuous network programs with arc delays. J. Inf. Optim. Sci. 10, 45–52 (1989)

    Google Scholar 

  4. Anderson E.J., Nash P.: Linear Programming in Infinite-Dimensional Spaces. Wiley, New York (1987)

    Google Scholar 

  5. Anderson E.J., Nash P., Perold A.F.: Some properties of a class of continuous linear programs. SIAM J. Control Optim. 21, 258–265 (1983)

    Article  Google Scholar 

  6. Anderson E.J., Nash P., Philpott A.B.: A class of continuous network flow problems. Math. Oper. Res. 7, 501–514 (1982)

    Article  Google Scholar 

  7. Anderson E.J., Philpott A.B.: A continuous-time network simplex algorithm. Networks 19, 395–425 (1989)

    Article  Google Scholar 

  8. Anderson E.J., Philpott A.B.: Optimisation of flows in networks over time. In: Kelly, F.P. (ed.) Probability, Statistics and Optimisation, Chapter 27, pp. 369–382. Wiley, New York (1994)

    Google Scholar 

  9. Aronson J.E.: A survey of dynamic network flows. Ann. Oper. Res. 20, 1–66 (1989)

    Article  Google Scholar 

  10. Fleischer L., Sethuraman J.: Efficient algorithms for separated continuous linear programs: the multi-commodity flow problem with holding costs and extensions. Math. Oper. Res. 30, 916–938 (2005)

    Article  Google Scholar 

  11. Fonoberova M.: Algorithms for finding optimal flows in dynamic networks. In: Rebennack, S., Pardalos, P.M., Pereira, M.V.F., Iliadis, N.A. (eds) Handbook of Power Systems II, Energy Systems, pp. 31–54. Springer, Berlin (2010)

    Chapter  Google Scholar 

  12. Ford L.R., Fulkerson D.R.: Constructing maximal dynamic flows from static flows. Oper. Res. 6, 419–433 (1958)

    Article  Google Scholar 

  13. Ford L.R., Fulkerson D.R.: Flows in Networks. Princeton University Press, Princeton (1962)

    Google Scholar 

  14. Guisewite G., Pardalos P.M.: Minimum concave cost network flow problems: applications, complexity, and algorithms. Ann. Oper. Res. 25, 75–100 (1990)

    Article  Google Scholar 

  15. Hashemi S.M., Mokarami S., Nasrabadi E.: Dynamic shortest path problems with time-varying costs. Optim. Lett. 4, 147–156 (2010)

    Article  Google Scholar 

  16. Klinz B., Woeginger G.J.: Minimum-cost dynamic flows: the series-parallel case. Networks 43, 153–162 (2004)

    Article  Google Scholar 

  17. Luo X., Bertsimas D.: A new algorithm for state-constrained separated continuous linear programs. SIAM J. Control Optim. 37, 177–210 (1998)

    Article  Google Scholar 

  18. Nasrabadi, E.: Dynamic Flows in Time-varying Networks. PhD thesis, Technische Universität Berlin (Germany) and Amirkabir University of Technology (Iran) (2009)

  19. Nasrabadi E., Hashemi S.M.: Minimum cost time-varying network flow problems. Optim. Methods Softw. 25, 429–447 (2010)

    Article  Google Scholar 

  20. Pardalos, P.M., Du, D. (eds): Network Design: Connectivity and Facilities Location, vol. 40 of DIMACS Series. American Mathematical Society, Providence (1998)

    Google Scholar 

  21. Philpott A.B.: Network programming in continuous time with node storage. In: Anderson, E.J., Philpott, A.B. (eds) Infinite Programming: Proceedings of an International Symposium on Infinite Dimensional Linear Programming, vol. 259 of Lecture notes in economics and mathematical systems, pp. 136–153. Springer, Berlin (1985)

    Google Scholar 

  22. Philpott A.B.: Continuous-time flows in networks. Math. Oper. Res. 15, 640–661 (1990)

    Article  Google Scholar 

  23. Philpott A.B., Craddock M.: An adaptive discretization algorithm for a class of continuous network programs. Networks 26, 1–11 (1995)

    Article  Google Scholar 

  24. Powell W.B., Jaillet P., Odoni A.: Stochastic and dynamic networks and routing. In: Ball, M.O., Magnanti, T.L., Monma, C.L., Nemhauser, G.L. (eds) Network Routing, vol. 8 of Handbooks in Operations Research and Management Science, chapter 3, pp. 141–295. North–Holland, Amsterdam (1995)

    Google Scholar 

  25. Pullan M.C.: An algorithm for a class of continuous linear programs. SIAM J. Control Optim. 31, 1558–1577 (1993)

    Article  Google Scholar 

  26. Pullan M.C.: Forms of optimal solutions for separated continuous linear programs. SIAM J. Control Optim. 33, 1952–1977 (1995)

    Article  Google Scholar 

  27. Pullan M.C.: A duality theory for separated continuous linear programs. SIAM J. Control Optim. 34, 931–965 (1996)

    Article  Google Scholar 

  28. Pullan M.C.: Existence and duality theory for separated continuous linear programs. Math. Model. Syst. 3, 219–245 (1997)

    Article  Google Scholar 

  29. Pullan M.C.: A study of general dynamic network programs with arc time-delays. SIAM J. Optim. 7, 889–912 (1997)

    Article  Google Scholar 

  30. Pullan M.C.: Convergence of a general class of algorithms for separated continuous linear programs. SIAM J. Optim. 10, 722–731 (2000)

    Article  Google Scholar 

  31. Skutella M.: An introduction to network flows over time. In: Cook, W., Lovász, L., Vygen, J. (eds) Research Trends in Combinatorial Optimization, pp. 451–482. Springer, Berlin (2009)

    Chapter  Google Scholar 

  32. Weiss G.: A simplex based algorithm to solve separated continuous linear programs. Math. Program. 115, 151–198 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ebrahim Nasrabadi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hashemi, S.M., Nasrabadi, E. On solving continuous-time dynamic network flows. J Glob Optim 53, 497–524 (2012). https://doi.org/10.1007/s10898-011-9723-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-011-9723-0

Keywords

Navigation