Mixed integer linear models for the optimization of dynamical transport networks

Abstract

We introduce a mixed integer linear modeling approach for the optimization of dynamic transport networks based on the piecewise linearization of nonlinear constraints and we show how to apply this method by two examples, transient gas and water supply network optimization. We state the mixed integer linear programs for both cases and provide numerical evidence for their suitability.

This is a preview of subscription content, access via your institution.

References

  1. Abreu J, Cabrera E, Izquierdo J, García-Serra J (1999) Flow modeling in pressurized systems revisited. J Hydraul Eng 125: 1154–1169

    Article  Google Scholar 

  2. Domschke P, Geiß B, Kolb O, Lang J, Martin A, Morsi A (2010) Combination of nonlinear and linear optimization of transient gas networks. INFORMS J comp. doi:10.1287/ijoc.1100.0429

  3. Ehrhardt K, Steinbach M (2003) Nonlinear optimization in gas networks. Technical report, Konrad- Zuse-Zentrum für Informationstechnik, Berlin

  4. Ehrhardt K, Steinbach M (2005) Nonlinear optimization in gas networks. In: Bock HG, Kostina E, Phu HX, Rannacher R (eds) Modeling, simulation and optimization of complex processes. Springer, Berlin, pp 139–148

    Google Scholar 

  5. Geißler B, Martin A, Morsi A, Schewe L (2010) Using piecewise linear functions for solving MINLPs. To appear in the IMA Volume on MINLP

  6. ILOG CPLEX Division (2002) 889 Alder Avenue, Suite 200, Incline Village, NV 89451, USA. Using the CPLEX Callable Library. Information available at url:http://www.cplex.com

  7. Kolb O, Lang J, Bales P (2010) An implicit box scheme for subsonic compressible flow with dissipative source term. Numer Algorithms 53(2):293–307. http://www.springerlink.com/content/p835883ku82m55k2/

    Google Scholar 

  8. Lee J, Margot J, Margot F (2004) Min-up/ min-down polytopes. Discret Optim 1: 77–85

    MathSciNet  MATH  Article  Google Scholar 

  9. LeVeque RJ (2002) Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics, Cambridge

    Google Scholar 

  10. Manne AS, Markowitz HM (1957) On the solution of discrete programming problems. Econometrica 25: 84–110

    MathSciNet  MATH  Article  Google Scholar 

  11. Möller M (2004) Mixed integer models for the optimisation of gas networks in the stationary case. PhD thesis, Darmstadt University of Technology

  12. Moritz S (2006) A mixed integer approach for the transient case of gas network optimization. PhD thesis, Darmstadt University of Technology

  13. Nemhauser GL, Wolsey LA (1988) Integer and combinatorial optimization. Wiley, New York

    Google Scholar 

  14. Padberg M (2000) Approximating separable nonlinear functions via mixed zero-one programs. Oper Res Lett 27: 1–5

    MathSciNet  MATH  Article  Google Scholar 

  15. Sekirnjak E (2000, Nov) Transiente technische optimierung (tto-prototyp). Technical report, PSI AG, Berlin

  16. Tomlin JA (1981) A suggested extension of special ordered sets to non-separable non-convex programming problems. Ann Discret Math 11: 359–370

    MathSciNet  MATH  Google Scholar 

  17. Wendroff B (1960) On centered difference equations for hyperbolic systems. SIAM J Soc Ind Appl Math 8(3): 549–555

    MathSciNet  MATH  Article  Google Scholar 

  18. Wilson D (1998) Polyhedral methods for piecewise linear Functions. PhD thesis, University of Kentucky

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Björn Geißler.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Geißler, B., Kolb, O., Lang, J. et al. Mixed integer linear models for the optimization of dynamical transport networks. Math Meth Oper Res 73, 339 (2011). https://doi.org/10.1007/s00186-011-0354-5

Download citation

Keywords

  • Mixed integer linear programming
  • Piecewise linear approximation
  • Gas network optimization
  • Water network optimization