Journal of Global Optimization

, Volume 68, Issue 3, pp 623–640 | Cite as

Approximation guarantees of algorithms for fractional optimization problems arising in dispatching rules for INDS problems

  • Hongtan Sun
  • Thomas C. Sharkey


In this paper, we provide approximation guarantees of algorithms for the fractional optimization problems arising in the dispatching rules from recent literature for Integrated Network Design and Scheduling problems. These fractional optimization problem are proved to be NP-hard. The approximation guarantees are based both on the number of arcs in the network and on the number of machines in the scheduling environment. We further demonstrate, by example, the tightness of the factors for these approximation algorithms.


Approximation algorithm Fractional optimization Integrated Network Design and Scheduling Complexity analysis 



The work of Thomas C. Sharkey was supported in part by the U.S. National Science Foundation under Grant Number CMMI-1254258.


  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall Inc, New Jersey (1993)MATHGoogle Scholar
  2. 2.
    Averbakh, I., Pereira, J.: The flowtime network construction problem. IIE Trans. 44, 681–694 (2012)CrossRefGoogle Scholar
  3. 3.
    Bajalinov, E.B.: Linear-Fractional Programming: Theory, Method, Applications and Software. Kluwer, Boston (2003)CrossRefMATHGoogle Scholar
  4. 4.
    Baxter, M., Elgindy, T., Ernst, A.T., Kalinowski, T., Savelsbergh, M.W.P.: Incremental network design with shortest paths. Eur. J. Oper. Res. 238, 675–684 (2014)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Borrero, J.S., Gillen, C., Prokopyev, O.A.: Fractional 0–1 programing: applications and algorithms. J. Glob. Optim. (2016). doi: 10.1007/s10898-016-0487-4 Google Scholar
  6. 6.
    Borrero, J.S., Gillen, C., Prokopyev, O.A.: A simple technique to improve linearized reformulations of fractional (hyperbolic) 0–1 programming problems. Oper. Res. Lett. 44, 479–486 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    DeBlasio, R., Tom, C.: standards for the smart grid. In: Proceedings of 2008 IEEE Energy 2030 Conference, pp. 1–7. Atlanta, GA (2008)Google Scholar
  8. 8.
    Fang, S., Gao, D.Y., Sheu, R., Xing, W.: Global optimization for a class of fractional programming problems. J. Glob. Optim. 45, 337–353 (2009)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Farhangi, H.: The path of the smart grid. IEEE Power Energy Mag. 8, 18–28 (2010)CrossRefGoogle Scholar
  10. 10.
    Floudas, C.A., Gounaris, C.E.: A review of recent advances in global optimization. J. Glob. Optim. 45, 3–38 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kalinowski, T., Matsypura, D., Savelsbergh, M.W.P.: Incremental network design with maximum flows. Eur. J. Oper. Res. 242, 51–62 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Lee, E.E., Mithchell, J.E., Wallace, W.A.: Restoration of services in interdependent infrastructure systems: a network flows approach. IEEE Trans. Syst. Man Cybern. C Appl. Rev. 37(6), 1303–1317 (2007)CrossRefGoogle Scholar
  13. 13.
    Mahmood, A., Aamir, M., Anis, M.I.: Design and implementation of AMR smart grid system. In: Proceedings of 2008 IEEE Electrical Power & Energy Conference, Vancouver, BC, pp. 1-6 (2008)Google Scholar
  14. 14.
    Momoh, J.A.: Smart grid design for efficient and flexible power networks operation and control. In: Proceedings of 2009 IEEE/PES Power Syst Conference Exposition Seattle, WA, pp. 1–8 (2009)Google Scholar
  15. 15.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, New York (1999)MATHGoogle Scholar
  16. 16.
    Nurre, S.G., Cavdaroglu, B., Mitchell, J.E., Sharkey, T.C., Wallace, W.A.: Restoring infrastructure systems: an integrated network design and scheduling problem. Eur. J. Oper. Res. 223(3), 794–806 (2012)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Nurre, S.G., Sharkey, T.C.: Integrated network design and scheduling problems with parallel identical machines: complexity results and dispatching rules. Networks 63(4), 306–326 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ouyang, M.: Review on modeling and simulation of interdependent critical infrastructure systems. Reliab. Eng. Syst. Saf. 121, 43–60 (2014)CrossRefGoogle Scholar
  19. 19.
    Pardalos, P.M., Phillips, A.T.: Global optimization of fractional programs. J. Glob. Optim. 1, 173–182 (1991)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Pinedo, M.L.: Scheduling: Theory, Algorithms, and Systems, 4th edn. Springer, New York (2012)CrossRefMATHGoogle Scholar
  21. 21.
    Pochet, Y., Warichet, F.: A tighter continuous time formulation for the cyclic scheduling of a mixed plant. Comput. Chem. Eng. 32(11), 2723–2744 (2008)CrossRefGoogle Scholar
  22. 22.
    Sauter, T., Lobashov, M.: End to end communication architecture for smart grids. IEEE Trans. Ind. Electron. 58(4), 1218–1228 (2011)CrossRefGoogle Scholar
  23. 23.
    Ursulenko, O., Butenko, S., Prokopyev, O.A.: A global optimization algorithm for solving the minimum multiple ratio spanning tree problem. J. Glob. Optim. 56, 1029–1043 (2013)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Wang, Y.J., Shen, P.P., Liang, Z.: A branch-and-bound algorithm to globally solve the sum of several linear ratios. Appl. Math. Comput. 168(1), 89–101 (2005)MathSciNetMATHGoogle Scholar
  25. 25.
    Warichet, F.: Scheduling of mixed batch-continuous production lines. PhD thesis, Université Catholique de Louvain, Belguim (2007)Google Scholar
  26. 26.
    Yue, D., Guilén-Gasálbez, G., You, F.: Global optimization of large-scale mixed-integer linear fractional programming problems: a reformulation-linearization method and process scheduling applications. AIChE J. 59, 4255–4272 (2013)CrossRefGoogle Scholar
  27. 27.
    You, F., Castro, P.M., Grossmann, I.E.: Dinkelbach’s algorithm as an efficient method to solve a class of MINLP models for large-scale cyclic scheduling problems. Comput. Chem. Eng. 33, 1879–1889 (2009)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Industrial and Systems EngineeringRensselaer Polytechnic InstituteTroyUSA

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