Skip to main content
Springer Nature Link
Log in

Tradeoff directions in multiobjective optimization problems

  • Published:
Mathematical Programming Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

Multiobjective optimization problems (MOP) typically have conflicting objectives wherein a gain in one objective is at the expense of another. Tradeoff directions, which measure the change in some objectives relative to changes in others, provide important information as to the best direction of improvement from the current solution. In this paper we present a general definition of tradeoffs as a cone of directions and provide a general method of calculating tradeoffs at every Pareto optimal point of a convex MOP. This extends current definitions of tradeoffs which assume certain conditions on the feasible set and the objective functions. Two comprehensive numerical examples are provided to illustrate the tradeoff directions and the methods used to calculate them.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. H.P. Benson, An improved definition of proper efficiency for vector maximization with respect to cones,Journal of Mathematical Analysis and Applications 71 (1979) 232–241.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. Borwein, Proper efficient points for maximization with respect to cones,SIAM Journal on Control and Optimization 15 (1977) 57–63.

    Article  MATH  MathSciNet  Google Scholar 

  3. J. Buchanan, Solution methods for multiple objective decision models, Ph.D. Thesis, University of Canterbury, New Zealand, 1985.

    Google Scholar 

  4. V. Chankong and Y.Y. Haimes, The interactive surrogate worth tradeoff (ISWT) method for multiobjective decision making, in: S. Zionts, ed.,Multiple Criteria Problem Solving, Lecture Notes in Economics and Mathematical Systems, Vol. 155 (Springer, New York, 1977) 42–67.

    Google Scholar 

  5. A.M. Geoffrion, Proper efficiency and the theory of vector maximization,Journal of Mathematical Analysis and Applications 22 (1968) 613–630.

    Article  MathSciNet  Google Scholar 

  6. A.M. Geoffrion, J.S. Dyer and A. Feinberg, An interactive approach for multicriterion optimization with application to the operation of an academic department,Management Science 19 (1972) 357–368.

    MATH  Google Scholar 

  7. Y.Y. Haimes, L. Lasdon and D. Wismer, On a bicriterion formulation of the problems of integrated systems identification and system optimization,IEEE Transactions on Systems Man and Cybernetics 1 (1971) 296–297.

    Article  MATH  MathSciNet  Google Scholar 

  8. Y.Y. Haimes and V. Chankong, Kuhn-Tucker multipliers as trade-offs in multiobjective decision-making analysis,Automatica 15 (1979) 59–72.

    Article  MATH  Google Scholar 

  9. M. Halme, Local characterization of efficient solutions in interactive multiple objective linear programming, Helsinki School of Economics and Business Administration, Helsinki, 1992.

    Google Scholar 

  10. M.I. Henig, A generalized method of approximating the set of efficient points with respect to a convex cone, in: J. Morse, ed.,Organizations: Multiple Agents with Multiple Criteria, Lecture Notes in Economics and Mathematical Systems (Springer, Berlin, 1981) 140–144.

    Google Scholar 

  11. M.I. Henig, Proper efficiency with respect to cones,Journal of Optimization Theory and Applications 36 (1982) 387–407.

    Article  MATH  MathSciNet  Google Scholar 

  12. H. Isermann, Proper efficiency and the linear vector maximization problem,Operations Research 22 (1974) 189–191.

    MATH  MathSciNet  Google Scholar 

  13. I. Kaliszewski,Qualitative Pareto Analysis by Cone Separation Technique (Kluwer Academic Publishers, Boston, MA, 1994).

    Google Scholar 

  14. H.W. Kuhn and A.W. Tucker, Nonlinear programming, in:Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability (University of California Press, Berkeley, CA, 1951) 481–492.

    Google Scholar 

  15. R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1972).

    Google Scholar 

  16. M. Sakawa and H. Yano, Trade-off rates in the hyperplane method for multi-objective optimization,European Journal of Operational Research 44 (1990) 105–118.

    Article  MATH  MathSciNet  Google Scholar 

  17. W.S. Shin and A. Ravindran, Interactive multiple objective optimization: Survey I — Continuous case,Computers and Operations Research 18 (1991) 97–114.

    Article  MATH  MathSciNet  Google Scholar 

  18. P.-L. Yu, Cone convexity, cone extreme points, and nondominated solutions, in decision problems with multiobjectives,Journal of Optimization Theory and Applications 14 (1974) 319–377.

    Article  MATH  MathSciNet  Google Scholar 

  19. S. Zionts and J. Wallenius, An interactive programming method for solving the multiple criteria problem,Management Science 29 (1976) 519–529.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Management, Tel Aviv University, Tel Aviv, Israel

    Mordechai I. Henig

  2. Department of Management Systems, University of Waikato, Hamilton, New Zealand

    John T. Buchanan

Authors
  1. Mordechai I. Henig
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. John T. Buchanan
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Henig, M.I., Buchanan, J.T. Tradeoff directions in multiobjective optimization problems. Mathematical Programming 78, 357–374 (1997). https://doi.org/10.1007/BF02614361

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02614361

Keywords