Skip to main content
SpringerLink
Log in

Outcome-based algorithm for optimizing over the efficient set of a bicriteria linear programming problem

  • Contributed Papers
  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This article presents a finite, outcome-based algorithm for optimizing a lower semicontinuous function over the efficient set of a bicriteria linear programming problem. The algorithm searches the efficient faces of the outcome set of the bicriteria linear programming problem. It exploits the fact that the dimension of the outcome set of the bicriteria problem is at most two. As a result, in comparison to decisionbased approaches, the number of efficient faces that need to be found is markedly reduced. Furthermore, the dimensions of the efficient faces found are always at most one. The algorithm can be implemented for a wide variety of lower semicontinuous objective functions.

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. Cohon, J. L.,Multiobjective Programming and Planning, Academic Press, New York, New York, 1978.

    Google Scholar 

  2. Evans, G. W.,An Overview of Techniques for Solving Multiobjective Mathematical Programs, Management Science, Vol. 30, pp. 1268–1282, 1984.

    Google Scholar 

  3. Goicoechea, A., Hansen, D. R., andDuckstein, L.,Multiobjective Decision Analysis with Engineering and Business Applications, John Wiley and Sons, New York, New York, 1982.

    Google Scholar 

  4. Kuhn, H. W., andTucker, A. W.,Nonlinear Programming, Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, Edited by J. Neyman, University of California Press, Berkeley, California, pp. 481–492, 1950.

    Google Scholar 

  5. Luc, D. T.,Theory of Vector Optimization, Springer Verlag, Berlin, Germany, 1989.

    Google Scholar 

  6. Ringuest, J. L.,Multiobjective Optimization: Behavioral and Computational Considerations, Kluwer Academic Publishers, Boston, Massachusetts, 1992.

    Google Scholar 

  7. Rosenthal, R. E.,Principles of Multiobjective Optimization, Decision Sciences, Vol. 16, pp. 133–152, 1985.

    Google Scholar 

  8. Sawaragi, Y., Nakayama, H., andTanino, T.,Theory of Multiobjective Optimization, Academic Press, Orlando, Florida, 1985.

    Google Scholar 

  9. Stadler, W.,A Survey of Multicriteria Optimization or the Vector Maximum Problem, Part 1: 1776–1960, Journal of Optimization Theory and Applications, Vol. 29, pp. 1–52, 1979.

    Google Scholar 

  10. Steuer, R. E.,Multiple Criteria Optimization: Theory, Computation, and Application, John Wiley and Sons, New York, New York, 1986.

    Google Scholar 

  11. Yu, P. L.,Multiple Criteria Decision Making, Plenum, New York, New York, 1985.

    Google Scholar 

  12. Yu, P. L.,Multiple Criteria Decision Making: Five Basic Concepts, Optimization, Edited by G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, North Holland, Amsterdam, Holland, pp. 663–699, 1989.

    Google Scholar 

  13. Zeleny, M.,Multiple Criteria Decision Making, McGraw-Hill, New York, New York, 1982.

    Google Scholar 

  14. Benson, H. P.,Optimization over the Efficient Set, Journal of Mathematical Analysis and Applications, Vol. 98, pp. 562–580, 1984.

    Google Scholar 

  15. Fulop, J.,A Cutting Plane Method for Linear Optimization over the Efficient Set, Generalized Convexity, Edited by S. Komlosi, T. Rapcsak, and S. Schaible, Springer Verlag, Berlin, Germany, pp. 374–385, 1994.

    Google Scholar 

  16. Thach, P. T., Konno, H., andYokota, D.,A Dual Approach to a Minimization on the Set of Pareto-Optimal Solutions, Working Paper, Institute of Human and Social Sciences, Tokyo Institute of Technology, Tokyo, Japan, 1994.

    Google Scholar 

  17. Philip, J.,Algorithms for the Vector Maximization Problem, Mathematical Programming, Vol. 2, pp. 207–229, 1972.

    Google Scholar 

  18. Gallagher, R. J., andSaleh, O. A.,A Representation of an Efficiency Equivalent Polyhedron for the Objective Set of a Multiple Objective Linear Program, European Journal of Operational Research (to appear).

  19. Shin, W. S., andRavindran, A.,Interactive Multiple Objective Optimization: Survey, Part 1: Continuous Case, Computers and Operations Research, Vol. 18, pp. 97–114, 1991.

    Google Scholar 

  20. Dessouky, M. I., Ghiassi, M., andDavis, W. J.,Estimates of the Minimum Nondominated Criterion Values in Multiple Criteria Decision Making, Engineering Costs and Production Economics, Vol. 10, pp. 95–104, 1986.

    Google Scholar 

  21. Isermann, H., andSteuer, R. E.,Computational Experience Concerning Payoff Tables and Minimum Criteria Values over the Efficient Set, European Journal of Operational Research, Vol. 33, pp. 91–97, 1987.

    Google Scholar 

  22. Reeves, G. R., andReid, R. C.,Minimum Values over the Efficient Set in Multiple Objective Decision Making, European Journal of Operational Research, Vol. 36, pp. 334–338, 1988.

    Google Scholar 

  23. Weistroffer, H. R.,Careful Use of Pessimistic Values Is Needed in Multiple Objectives Optimization, Operations Research Letters, Vol. 4, pp. 23–25, 1985.

    Google Scholar 

  24. Benson, H. P.,An All-Linear Programming Relaxation Algorithm for Optimization over the Efficient Set, Journal of Global Optimization, Vol. 1, pp. 83–104, 1991.

    Google Scholar 

  25. Benayoun, R., De Montgolfier, J., Tergny, J., andLaritchev, O.,Linear Programming with Multiple Objective Functions: Step Method (STEM), Mathematical Programming, Vol. 1, pp. 366–375, 1971.

    Google Scholar 

  26. Benson, H. P., Lee, D., andMcClure, J. P.,Applying Multiple Criteria Decision Making in Practice: The Citrus Rootstock Selection Problem in Florida, Discussion Paper, Department of Decision and Information Sciences, University of Florida, Gainesville, Florida, 1992.

    Google Scholar 

  27. Belenson, S., andKapur, K. C.,An Algorithm for Solving Multicriterion Linear Programming Problems with Examples, Operational Research Quarterly, Vol. 24, pp. 65–77, 1973.

    Google Scholar 

  28. Kok, M., andLootsma, F. A.,Pairwise-Comparison Methods in Multiple Objective Programming, with Applications in a Long-Term Energy-Planning Model, European Journal of Operational Research, Vol. 22, pp. 44–55, 1985.

    Google Scholar 

  29. Benson, H. P.,Concave Minimization: Theory, Applications, and Algorithms, Handbook of Global Optimization, Edited by R. Horst and P. Pardalos, Kluwer Academic Publishers, Dordrecht, Holland, pp. 43–148, 1995.

    Google Scholar 

  30. Horst, R.,Deterministic Global Optimization: Recent Advances and New Fields of Application, Naval Research Logistics, Vol. 37, pp. 433–471, 1990.

    Google Scholar 

  31. Horst, R., andTuy, H.,Global Optimization: Deterministic Approaches, 2nd Edition, Springer Verlag, Berlin, Germany, 1993.

    Google Scholar 

  32. Horst, R., andPardalos, P., Editors,Handbook of Global Optimization, Kluwer Academic Publishers, Dordrecht, Holland, 1995.

    Google Scholar 

  33. Bolintineanu, S.,Optimality Conditions for Minimization over the (Weakly or Properly) Efficient Set, Journal of Mathematical Analysis and Applications, Vol. 173, pp. 523–541, 1993.

    Google Scholar 

  34. Bolintineanu, S.,Minimization of a Quasi-concave Function over an Efficient Set, Mathematical Programming, Vol. 61, pp. 89–110, 1993.

    Google Scholar 

  35. Benson, H. P.,A Finite, Nonadjacent Extreme Point Search Algorithm for Optimization over the Efficient Set, Journal of Optimization Theory and Applications, Vol. 73, pp. 47–64, 1992.

    Google Scholar 

  36. Benson, H. P.,A Bisection Extreme Point Search Algorithm for Optimizing over the Efficient Set in the Linear Dependence Case, Journal of Global Optimization, Vol. 3, pp. 95–111, 1993.

    Google Scholar 

  37. Benson, H. P., andSayin, S.,Optimization over the Efficient Set: Four Special Cases, Journal of Optimization Theory and Applications, Vol. 80. pp. 3–18, 1994.

    Google Scholar 

  38. Benson, H. P.,An Algorithm for Optimizing over the Weakly-Efficient Set, European Journal of Operational Research, Vol. 25, pp. 192–199, 1986.

    Google Scholar 

  39. Ecker, J. G., andSong, J. H.,Optimizing a Linear Function over an Efficient Set, Working Paper, Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York, 1993.

    Google Scholar 

  40. Muu, L. P.,A Method for Optimizing a Linear Function over the Efficient Set, Working Paper, Institute of Mathematics, Hanoi, Vietnam, 1992.

    Google Scholar 

  41. Dauer, J. P.,Optimization over the Efficient Set Using an Active Constraint Approach, Zeitschrift für Operations Research, Vol. 35, pp. 185–195, 1991.

    Google Scholar 

  42. Benson, H. P., andSayin, S.,A Face Search Heuristic for Optimizing over the Efficient Set, Naval Research Logistics, Vol. 40, pp. 103–116, 1993.

    Google Scholar 

  43. Korhonen, P., Salo, S., andSteuer, R.,A Heuristic for Estimating Nadir Criterion Values in Multiple Objective Linear Programming, Working Paper, Helsinki School of Economics and Business Administration, Helsinki, Finland, 1992.

    Google Scholar 

  44. Aksoy, Y.,An Interactive Branch-and-Bound Algorithm for Bicriterion Nonconvex/Mixed Integer Programming, Naval Research Logistics, Vol. 37, pp. 403–417, 1990.

    Google Scholar 

  45. Aneja, Y. P., andNair, K. P. K.,Bicriteria Transportation Problem, Management Science, Vol. 25, pp. 73–78, 1979.

    Google Scholar 

  46. Benson, H. P.,Vector Maximization with Two Objective Functions, Journal of Optimization Theory and Applications, Vol. 28, pp. 253–257, 1979.

    Google Scholar 

  47. Benson, H. P., andMorin, T. L.,A Bicriteria Mathematical Programming Model for Nutrition Planning in Developing Nations, Management Science, Vol. 33, pp. 1593–1601, 1987.

    Google Scholar 

  48. Cohon, J. L., Church, R. L., andSheer, D. P.,Generating Multiobjective Tradeoffs: An Algorithm for Bicriterion Problems, Water Resources Research, Vol. 15, pp. 1001–1010, 1979.

    Google Scholar 

  49. Gearhart, W. B.,On the Characterization of Pareto-Optimal Solutions in Bicriteria Optimization, Journal of Optimization Theory and Applications Vol. 27, pp. 301–307, 1979.

    Google Scholar 

  50. Geoffrion, A. M.,Solving Bicriterion Mathematical Programs, Operations Research, Vol. 15, pp. 39–54, 1967.

    Google Scholar 

  51. Kiziltan, G., andYucaoglu, E.,An Algorithm for Bicriterion Linear Programming, European Journal of Operational Research, Vol. 10, pp. 406–411, 1982.

    Google Scholar 

  52. Klein, G., Moskowitz, H., andRavindran, A.,Comparative Evaluation of Prior versus Progressive Articulation of Preferences in Bicriterion Optimization, Naval Research Logistics, Vol. 33, pp. 309–323, 1986.

    Google Scholar 

  53. Shin, W. S., andAllen, D. B.,An Interactive Paired Comparison Method for Bicriterion Integer Programming, Naval Research Logistics, Vol. 41, pp. 423–434, 1994.

    Google Scholar 

  54. Shin, W. S., andLee, J. J.,A Multirun Interactive Method for Bicriterion Optimization Problems, Naval Research Logistics, Vol. 39, pp. 115–135, 1992.

    Google Scholar 

  55. Walker, J.,An Interactive Method as an Aid in Solving Bicriterion Mathematical Programming Problems, Journal of the Operational Research Society, Vol. 29, pp. 915–922, 1978.

    Google Scholar 

  56. Haimes, Y. Y., Wismer, D. A., andLasdon, L. S.,On the Bicriterion Formulation of Integrated System Identification and Systems Optimization, IEEE Transactions on Systems, Man, and Cybernetics, Vol. 1, pp. 296–297, 1971.

    Google Scholar 

  57. Cohon, J. L., Scavone, G., andSolanki, R.,Multicriterion Optimization in Resources Planning, Multicriteria Optimization in Engineering and in the Sciences, Edited by W. Stadler, Plenum Press, New York, New York, pp. 117–160, 1988.

    Google Scholar 

  58. Van Wassenhove, L. N., andGelders, L. F.,Solving a Bicriterion Scheduling Problem, European Journal of Operational Research, Vol. 4, pp. 42–48, 1980.

    Google Scholar 

  59. Dauer, J. P.,Analysis of the Objective Space in Multiple Objective Linear Programming, Journal of Mathematical Analysis and Applications Vol. 126, pp. 579–593, 1987.

    Google Scholar 

  60. Dauer, J. P.,On Degeneracy and Collapsing in the Construction of the Set of Objective Values in a Multiple Objective Linear Program, Annals of Operations Research, Vol. 47, pp. 279–292, 1993.

    Google Scholar 

  61. Dauer, J. P., andLiu, Y. H.,Solving Multiple Objective Linear Programs in Objective Space, European Journal of Operational Research, Vol. 46, pp. 350–357, 1990.

    Google Scholar 

  62. Dauer, J. P., andSaleh, O. A.,Constructing the Set of Efficient Objective Values in Multiple Objective Linear Programs, European Journal of Operational Research, Vol. 46, pp. 358–365, 1990.

    Google Scholar 

  63. Benson, H. P.,A Geometrical Analysis of the Efficient Outcome Set in Multiple-Objective Convex Programs with Linear Criterion Functions, Journal of Global Optimization, Vol. 6, pp. 231–251, 1995.

    Google Scholar 

  64. Managasarian, O. L.,Nonlinear Programming, McGraw-Hill Book Company, New York, New York, 1969.

    Google Scholar 

  65. Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.

    Google Scholar 

  66. Arrow, K. J., Barankin, E. W., andBlackwell, D.,Admissible Points of Convex Sets, Contributions to the Theory of Games, Edited by H. W. Kuhn and A. W. Tucker, Princeton University Press, Princeton, New Jersey, pp. 87–91, 1953.

    Google Scholar 

  67. Bitran, G. R., andMagnanti, T. L.,The Structure of Admissible Points with Respect to Cone Dominance, Journal of Optimization Theory and Applications, Vol. 29, pp. 573–614, 1979.

    Google Scholar 

  68. Benson, H. P.,Admissible Points of a Convex Polyhedron, Journal of Optimization Theory and Applications, Vol. 38, pp. 341–361, 1982.

    Google Scholar 

  69. Blackwell, D., andGirshick, M. A.,Theory of Games and Statistical Decisions, Dover Publications, New York, New York, 1954.

    Google Scholar 

  70. Murty, K. G.,Linear Programming, John Wiley and Sons, New York, New York, 1983.

    Google Scholar 

  71. Benson, H. P.,Complete Efficiency and the Initialization of Algorithms for Multiple Objective Programming, Operations Research Letters, Vol. 10, pp. 481–487, 1991.

    Google Scholar 

  72. Bazaraa, M. S., Jarvis, J. J., andSherali, H. D.,Linear Programming and Network Flows, 2nd Edition, John Wiley and Sons, New York, New York, 1990.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Business Administration, University of Florida, Gainesville, Florida

    H. P. Benson (Professor)

  2. College of Business Administration, Myung Ji University, Seoul, Korea

    D. Lee (Lecturer)

Authors
  1. H. P. Benson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Lee
    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

Benson, H.P., Lee, D. Outcome-based algorithm for optimizing over the efficient set of a bicriteria linear programming problem. J Optim Theory Appl 88, 77–105 (1996). https://doi.org/10.1007/BF02192023

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02192023

Key Words

Search

Navigation