Computational Optimization and Applications

, Volume 28, Issue 3, pp 357–372 | Cite as

A Post-Optimality Analysis Algorithm for Multi-Objective Optimization

  • Vandana Venkat
  • Sheldon H. Jacobson
  • James A. Stori
Article

Abstract

Algorithms for multi-objective optimization problems are designed to generate a single Pareto optimum (non-dominated solution) or a set of Pareto optima that reflect the preferences of the decision-maker. If a set of Pareto optima are generated, then it is useful for the decision-maker to be able to obtain a small set of preferred Pareto optima using an unbiased technique of filtering solutions. This suggests the need for an efficient selection procedure to identify such a preferred subset that reflects the preferences of the decision-maker with respect to the objective functions. Selection procedures typically use a value function or a scalarizing function to express preferences among objective functions. This paper introduces and analyzes the Greedy Reduction (GR) algorithm for obtaining subsets of Pareto optima from large solution sets in multi-objective optimization. Selection of these subsets is based on maximizing a scalarizing function of the vector of percentile ordinal rankings of the Pareto optima within the larger set. A proof of optimality of the GR algorithm that relies on the non-dominated property of the vector of percentile ordinal rankings is provided. The GR algorithm executes in linear time in the worst case. The GR algorithm is illustrated on sets of Pareto optima obtained from five interactive methods for multi-objective optimization and three non-linear multi-objective test problems. These results suggest that the GR algorithm provides an efficient way to identify subsets of preferred Pareto optima from larger sets.

pareto optimality post-optimality analysis multi-objective optimization greedy reduction algorithm percentile levels 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    B.D. Craven, “On sensitivity analysis for multicriteria optimization,” Optimization, vol. 19, no. 4, pp. 513-523, 1988.Google Scholar
  2. 2.
    J. Dauer and Y.H. Liu, “Multi-criteria and goal programming,” in Advances in Sensitivity Analysis and Parametric Programming. Kluwer Academic Publishers: Boston, 1997, pp. 11-1-11-31.Google Scholar
  3. 3.
    D. Eby, R.C. Averill, W.F. Punch III, and E. D. Goodman, “The optimization of flywheels using an injection island genetic algorithm,” in Evolutionary Design By Computers, P. Bentley (Ed.), Morgan Kauffman Publishers, 1999.Google Scholar
  4. 4.
    H.A. Eiselt, P. Pederzoli, and C.L. Sandblom, “Continuous optimization models,” Operations Research: Theory, Techniques, Applications, Walter de Gruyter & Co., Berlin, 1987.Google Scholar
  5. 5.
    A.V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Mathematics in Science and Engineering, Academic Press Inc.: New York, 1983, vol. 165.Google Scholar
  6. 6.
    K. Fujita, N. Hirokawa, S. Akagi, S. Kitamura, and H. Yokohata, “Multi-objective optimal design of automotive engine using genetic algorithm,” in Proc. of ASME Design Engineering Technical Conferences, Atlanta, Georgia, 1998.Google Scholar
  7. 7.
    T. Gal, Post Optimal Analyses, Parametric Programming and Related Topics, Walter de Gruyter & Co.: New York, 1995.Google Scholar
  8. 8.
    T. Gal and K. Wolf, “Stability in vector optimization-A survey,” European Journal of Operations Research, vol. 25, no. 2, pp. 169-182, 1986.Google Scholar
  9. 9.
    D.R. Insua, Sensitivity Analysis in Multi-Objective Decision Making, Lecture Notes in Economics and Mathematical Systems, Springer-Verlag: New York, 1990.Google Scholar
  10. 10.
    R.L. Keeney and H. Raiffa, Decisions with Multiple Objectives: Preferences and Value Tradeoffs, John Wiley & Sons, Inc.: New York, 1976.Google Scholar
  11. 11.
    P. Korhonen and M. Halme, “Supporting the decision maker to find the most preferred solutions for a MOLP-problem,” in Proc. of the 9th Int. Conf. on Multiple Criteria Decision Making, Fairfax, Virginia, USA, 1990, pp. 173-183.Google Scholar
  12. 12.
    T.J. Lowe, J.F. Thisse, J.E. Ward, and R.E. Wendell, “On efficient solutions to multiple objective mathematical programs,” Management Science, vol. 30, no. 11, pp. 1346-1349, 1984.Google Scholar
  13. 13.
    D.T. Luc, Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems 319, Springer-Verlag: Berlin, Heidelberg, 1989.Google Scholar
  14. 14.
    H. Nakayama, “Trade-off analysis based upon parametric optimization,” in Proc. of the Intl. Workshop on Multiple Criteria Decision Support, Helsinki, Finland, 1989, pp. 42-52.Google Scholar
  15. 15.
    S.C. Narula, L. Kirilov, and V. Vassilev, “An interactive algorithm for solving multiple objective nonlinear programming problems,” in Proc. of the 10th Int. Joint Conf. on Multiple Criteria Decision Making, Taipei, Taiwan, 1983, pp. 119-127.Google Scholar
  16. 16.
    H.M. Rarig and Y.Y. Haimes, “Risk/dispersion index method,” IEEE Transactions on Systems, Man and Cybernetics, vol. 13, no. 3, pp. 317-328, 1983.Google Scholar
  17. 17.
    Y. Sawaragi, H. Nakayama, and T. Tanino, Theory of Multiobjective Optimization, Academic Press, Inc., Orlando, FL, 1985.Google Scholar
  18. 18.
    J. Siskos and N. Assimakopoulos, “Multicriteria highway planning: A case study,” Models and Methods in Multiple Criteria decision-making, Pergamon Press: Oxford, United Kingdom, 1989.Google Scholar
  19. 19.
    R.E. Steuer, Multiple Criteria Optimization: Theory, Computation and Application, John Wiley & Sons, Inc.: New York, 1986.Google Scholar
  20. 20.
    T. Tanino, “Sensitivity analysis in multiobjective optimization,” Journal of Optimization Theory and Applications, vol. 56, no. 1, pp. 479-499, 1988.Google Scholar
  21. 21.
    T. Tanino, “Stability and sensitivity analysis in convex vector optimization,” SIAM Journal on Control and Optimizationvol, vol. 26, no. 3, pp. 521-536, 1988.Google Scholar
  22. 22.
    T. Tanino, “Stability and sensitivity analysis in multiobjective nonlinear programming,” Annals of Operations Research, vol. 27, no. 1-4, pp. 97-114, 1990.Google Scholar
  23. 23.
    T. Tanino and Y. Sawaragi, “Stability of nondominated solutions in multicriteria decision-making,” Journal of Optimization Theory and Applications, vol. 30, no. 2, pp. 229-253, 1980.Google Scholar
  24. 24.
    D.A. Van Veldhuizen, “Multiobjective evolutionary algorithms: Classifications, analyses, and new innovations,” Ph.D. thesis, Department of Electrical and Computer Engineering, Graduate School of Engineering, Air Force Institute of Technology, Wright-Patterson AFB, Ohio, 1999.Google Scholar
  25. 25.
    V. Venkat, J.A. Stori, and S.H. Jacobson, “A comparison of interactive methods for multi-objective optimization: A computational study,” Technical Report, Simulation and Optimization Laboratory, University of Illinois, Urbana, IL, May 2004.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Vandana Venkat
    • 1
  • Sheldon H. Jacobson
    • 1
  • James A. Stori
    • 1
  1. 1.Department of Mechanical and Industrial EngineeringUniversity of IllinoisUrbanaUSA

Personalised recommendations