An interactive method for multiple-objective mathematical programming problems

  • W. S. Shin
  • A. Ravindran
Contributed Papers


An interactive method is developed for solving the general nonlinear multiple objective mathematical programming problems. The method asks the decision maker to provide partial information (local tradeoff ratios) about his utility (preference) function at each iteration. Using the information, the method generates an efficient solution and presents it to the decision maker. In so doing, the best compromise solution is sought in a finite number of iterations. This method differs from the existing feasible direction methods in that (i) it allows the decision maker to consider only efficient solutions throughout, (ii) the requirement of line search is optional, and (iii) it solves the problems with linear objective functions and linear utility function in one iteration. Using various problems selected from the literature, five line search variations of the method are tested and compared to one another. The nonexisting decision maker is simulated using three different recognition levels, and their impact on the method is also investigated.

Key Words

Multiple-objective optimization mathematical programming interactive methods tradeoff cutting plane 


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  1. 1.
    Evans, G. W.,An Overview of Techniques Solving Multiobjective Mathematical Programs, Management Science, Vol. 30, pp. 1268–1282, 1984.Google Scholar
  2. 2.
    Shin, W. S., andRavindran, A.,Interactive MOMP Methods: A Survey, Working Paper No. 87-15, School of Industrial Engineering, University of Oklahoma, Norman, Oklahoma, 1987.Google Scholar
  3. 3.
    Klein, G., Moskowitz, H., andRavindran, A.,Comparative Evaluation of Prior versus Progressive Articulation of Preference in Bicriterion Optimization, Naval Research Logistics Quarterly, Vol. 33, pp. 309–323, 1986.Google Scholar
  4. 4.
    Rosenthal, R. E.,Concepts, Theory, and Techniques: Principles of Multiobjective Optimization, Decision Sciences, Vol. 16, pp. 133–152, 1985.Google Scholar
  5. 5.
    Zionts, S.,A Survey of Multiple-Criteria Integer Programming Methods, Annals of Discrete Mathematics, Vol. 5, pp. 389–398, 1979.Google Scholar
  6. 6.
    Chankong, V., andHaims, Y. Y.,Multiobjective Decision Making: Theory and Methodology, North-Holland, Amsterdam, Holland, 1983.Google Scholar
  7. 7.
    Hwang, C. L., andMasud, A. S. M.,Multiple-Objective Decision Making: Methods and Applications, Springer-Verlag, New York, New York, 1979.Google Scholar
  8. 8.
    Steuer, R. E.,Multiple-Objective Optimization: Theory, Computation, and Application, John Wiley and Sons, New York, New York, 1986.Google Scholar
  9. 9.
    Zeleny, M.,Multiple-Criteria Decision Making, McGraw-Hill, New York, New York, 1982.Google Scholar
  10. 10.
    Geoffrion, A. M., Dyer, J. S., andFeinberg, A.,An Interactive Approach for Multicriterion Optimization with an Application to the Operation of an Academic Department, Management Science, Vol. 19, pp. 357–368, 1972.Google Scholar
  11. 11.
    Dyer, J. S.,A Time-Sharing Computer Program for the Solution of the Multiple-Criteria Problem, Management Science, Vol. 19, pp. 1379–1382, 1973.Google Scholar
  12. 12.
    Oppenheimer, K. R.,A Proxy Approach to Multi-Attribute Decision Making, Management Science, Vol. 24, pp. 675–689, 1978.Google Scholar
  13. 13.
    Rosinger, E. E.,Interactive Algorithm for Multiobjective Optimization, Journal of Optimization Theory and Applications, Vol. 35, pp. 339–365, 1981.Google Scholar
  14. 14.
    Hemming, T.,Some Modifications of a Large-Step Gradient Method for Interactive Multicriterion Optimization, Organizations: Multiple Agents with Multiple Criteria, Edited by J. N. Morse, Springer-Verlag, New York, New York, 1981.Google Scholar
  15. 15.
    Musselman, K., andTalavage, J.,A Tradeoff Cut Approach to Multiple-Objective Optimizations, Operations Research, Vol. 28, pp. 1424–1435, 1980.Google Scholar
  16. 16.
    Loganathan, G. V., andSherali, H. D.,A Convergent Interactive Cutting Plane Algorithm for Multiobjective Optimization, Operations Research, Vol. 35, pp. 365–377, 1987.Google Scholar
  17. 17.
    Sadagopan, S., andRavindran, A.,Interactive Algorithms for Multi-Criteria Nonlinear Programming Problems, European Journal of Operations Research, Vol. 25, pp. 247–257, 1986.Google Scholar
  18. 18.
    Geoffrion, A. M.,Proper Efficiency and the Theory of Vector Maximization, Journal of Mathematical Analysis and Applications, Vol. 22, pp. 618–630, 1968.Google Scholar
  19. 19.
    Yu, P. L.,Cone Convexity, Cone Extreme Points, and Nondominated Solutions in Decision Problems with Multiple Objectives, Journal of Optimization Theory and Applications, Vol. 14, pp. 319–377, 1974.Google Scholar
  20. 20.
    Steuer, R. E.,An Interactive Multiple-Objective Linear Programming Procedures, TIMS Studies in the Management Sciences, Vol. 6, pp. 225–239, 1977.Google Scholar
  21. 21.
    Zionts, S., andWallenius, J.,An Interactive Programming Method for Solving Multiple-Criteria Problems, Management Science, Vol. 22, pp. 652–663, 1976.Google Scholar
  22. 22.
    Abadie, J., andCarpenter, J.,Generalization of the Wolfes' Reduced Gradient Method to the Case of Nonlinear Constraints, Optimization, Edited by R. Fletcher, Academic Press, New York, New York, pp. 37–48, 1969.Google Scholar
  23. 23.
    Lasdon, L. S., andWaren, A. D.,GRG2 User's Guide, Unpublished Manuscript, 1983.Google Scholar
  24. 24.
    Ringuest, J. L., andGulledge, T. R. Jr.,Interactive Multiobjective Complex Search, European Journal of Operations Research, Vol. 19, pp. 362–371, 1985.Google Scholar
  25. 25.
    Walker, J.,An Interactive Method as an Aid in Solving Bicriterion Mathematical Programming Problems, Journal of Operational Research Society, Vol. 29, pp. 915–922, 1978.Google Scholar
  26. 26.
    Sakawa, M.,An Interactive Computer Program for Multiobjective Decision Making by the Sequential Proxy Optimization Technique (SPOT), European Journal of Operations Research, Vol. 9, pp. 386–398, 1982.Google Scholar
  27. 27.
    Zionts, S., andWallenius, J.,An Interactive Multi-Objective Linear Programming Method for a Class of Underlying Nonlinear Utility Functions, Management Science, Vol. 29, pp. 519–529, 1983.Google Scholar
  28. 28.
    Musselman, K.,An Interactive Tradeoff Cutting Plane Approach to Continuous and Discrete Multiple-Objective Optimization, Ph.D. Thesis, Purdue University, 1978.Google Scholar

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • W. S. Shin
    • 1
  • A. Ravindran
    • 2
  1. 1.Department of Industrial EngineeringMississippi State University
  2. 2.University of OklahomaNorman

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