Skip to main content
SpringerLink
Log in

Multiobjective integer nonlinear fractional programming problem: A cutting plane approach

  • Theoretical Article
  • Published:
OPSEARCH Aims and scope Submit manuscript

Abstract

The present paper discusses a multiobjective integer nonlinear fractional programming problem based on cutting plane technique. The methodology discussed is such that it finds all the nondominated t-tuples of the multiobjective nonlinear fractional programming problem by exploiting the quasimonotone character of the nonlinear fractional functions involved. The cut discussed in the present paper scans and truncates a portion of the feasible region in such way that once truncated, it does not reappear, thereby leading to the convergence of the proposed algorithm in finite number of steps. Further, the quasimonotone character of the objective functions involved enables us to find all the nondominated t-tuples at extreme points of the truncated feasible region obtained after repeated applications of the cut developed in the paper.

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

Access this article

Log in via an institution

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. Alves, M.J., Clímaco, J.: A review of interactive methods for multiobjective integer and mixed integer programming. Eur. J. Oper. Res. 180, 99–115 (2007)

    Article  Google Scholar 

  2. Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms. Wiley, New York (1991)

    Google Scholar 

  3. Caballero, R., Hernández, M.: The controlled estimation method in the multiobjective linear fractional problem. Comput. Oper. Res. 31, 1821–1832 (2004)

    Article  Google Scholar 

  4. Charles, V., Udhayakumar, A., Uthariaraj V.R.: An approach to find redundant objective function(s) and redundant constraint(s) in multiobjective nonlinear fractional programming problems. Eur. J. Oper. Res. 201, 390–398 (2010)

    Article  Google Scholar 

  5. Dahiya, K., Verma, V.: Valid cuts in integer programming with bounded variables. Proc. APORS 1, 19–28 (2003)

    Google Scholar 

  6. Dhaenens, C., Lemesre, J., Talbi, E.G.: K-PPM: a new exact method to solve multi-objective combinatorial optimization problems. Eur. J. Oper. Res. 200(1), 45–53 (2010)

    Article  Google Scholar 

  7. Ecker, J.G., Kouda, I.A.: Finding all extreme points for multiobjective linear programs. Math. Prog. 14, 249–261 (1978)

    Article  Google Scholar 

  8. Ehrgott, M., Gandibleux, X.: A survey and annotated bibliography of multiobjective combinatorial optimization. OR Spektrum 22(4), 425–460 (2000)

    Google Scholar 

  9. Evans, J.P., Steuer, R.E.: A revised simplex method for linear multiple objective programs. Math. Program. 5, 54–72 (1973)

    Article  Google Scholar 

  10. Geoffrion, A.M.: Solving bicriterion mathematical programs. Oper. Res. 15(1), 39–54 (1967)

    Article  Google Scholar 

  11. Gupta, R., Puri, M.C.: Bicriteria integer nonlinear fractional programming problem. Asia-Pac. J. Oper. Res. 15, 1–16 (1998)

    Google Scholar 

  12. Kornbluth, J.S.H., Steuer, R.E.: Multiple objective linear fractional programming. Manage. Sci. 27(9), 1024–1039 (1981)

    Article  Google Scholar 

  13. Jorge, J.M.: An algorithm for optimizing a linear function over an integer efficient set. Eur. J. Oper. Res. 195(1), 98–103, (2009)

    Article  Google Scholar 

  14. Metev, B., Gueorguieva, D.: A simple method for obtaining weakly efficient points in multiobjective linear fractional programming problems. Eur. J. Oper. Res. 126, 386–390 (2000)

    Article  Google Scholar 

  15. Özlen, M., Azizoǧlu, M.: Multi-objective integer programming: a general approach for generating all non-dominated solutions. Eur. J. Oper. Res. 199(1), 25–35 (2009)

    Article  Google Scholar 

  16. Przybylski, A., Gandibleux, X., Ehrgott, M.: A two phase method for multi-objective integer programming and its application to the assignment problem with three objectives. Discrete Optim. 7(3), 149–165 (2010)

    Article  Google Scholar 

  17. Saad, O.M., Biltagy, M.Sh., Farag, T.B.: An algorithm for multiobjective integer nonlinear fractional programming problem under fuzziness. Gen Math. Notes 2, 1–17 (2011)

    Google Scholar 

  18. Schaible, S.: Fractional programming: applications and algorithms. Eur. J. Oper. Res. 7, 111–120 (1981)

    Article  Google Scholar 

  19. Schaible S.: Fractional programming. In: Horst, R., Paradalos, P.M. (eds.) Handbook of Global Optimization, pp. 495–608. Kluwer Academic publishers, Dordrecht (1995)

    Google Scholar 

  20. Sniedovich M.: Fractional programming revisited. Eur. J. Oper. Res. 33, 334–341 (1988)

    Article  Google Scholar 

  21. Steuer, R.E.: Multiple Criteria Optimization - Theory, Computations and Application. Wiley, New York (1986)

    Google Scholar 

  22. Sylva, J., Crema, A.: A method finding the set of non-dominated vectors for multiple objective integer linear programs. Eur. J. Oper. Res. 158, 46–55 (2004)

    Article  Google Scholar 

  23. Zionts, S.: A survey of multiple criteria integer programming methods. Ann. Discrete Math. 1, 551–562 (1979)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Computer Applications (SMCA), Thapar University, Patiala, 147004, Punjab, India

    Vikas Sharma

Authors
  1. Vikas Sharma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vikas Sharma.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sharma, V. Multiobjective integer nonlinear fractional programming problem: A cutting plane approach. OPSEARCH 49, 133–153 (2012). https://doi.org/10.1007/s12597-012-0067-4

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12597-012-0067-4

Keywords

Search

Navigation