Introduction to Multiobjective Optimization: Noninteractive Approaches

  • Kaisa Miettinen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5252)


We give an introduction to nonlinear multiobjective optimization by covering some basic concepts as well as outlines of some methods. Because Pareto optimal solutions cannot be ordered completely, we need extra preference information coming from a decision maker to be able to select the most preferred solution for a problem involving multiple conflicting objectives. Multiobjective optimization methods are often classified according to the role of a decision maker in the solution process. In this chapter, we concentrate on noninteractive methods where the decision maker either is not involved or specifies preference information before or after the actual solution process. In other words, the decision maker is not assumed to devote too much time in the solution process.


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  1. Ballestero, E., Romero, C.: A theorem connecting utility function optimization and compromise programming. Operations Research Letters 10(7), 421–427 (1991)MathSciNetCrossRefMATHGoogle Scholar
  2. Benayoun, R., de Montgolfier, J., Tergny, J., Laritchev, O.: Programming with multiple objective functions: Step method (STEM). Mathematical Programming 1(3), 366–375 (1971)MathSciNetCrossRefMATHGoogle Scholar
  3. Benson, H.P.: Existence of efficient solutions for vector maximization problems. Journal of Optimization Theory and Application 26(4), 569–580 (1978)MathSciNetCrossRefMATHGoogle Scholar
  4. Benson, H.P.: Vector maximization with two objective functions. Journal of Optimization Theory and Applications 28(3), 253–257 (1979)MathSciNetCrossRefMATHGoogle Scholar
  5. Censor, Y.: Pareto optimality in multiobjective problems. Applied Mathematics and Optimization 4(1), 41–59 (1977)MathSciNetCrossRefMATHGoogle Scholar
  6. Chankong, V., Haimes, Y.Y.: Multiobjective Decision Making: Theory and Methodology. Elsevier Science Publishing, New York (1983)MATHGoogle Scholar
  7. Charnes, A., Cooper, W.W.: Management Models and Industrial Applications of Linear Programming, vol. 1. Wiley, New York (1961)MATHGoogle Scholar
  8. Charnes, A., Cooper, W.W.: Goal programming and multiple objective optimization; part 1. European Journal of Operational Research 1(1), 39–54 (1977)MathSciNetCrossRefMATHGoogle Scholar
  9. Charnes, A., Cooper, W.W., Ferguson, R.O.: Optimal estimation of executive compensation by linear programming. Management Science 1(2), 138–151 (1955)MathSciNetCrossRefMATHGoogle Scholar
  10. Cohon, J.L.: Multicriteria programming: Brief review and application. In: Gero, J.S. (ed.) Design Optimization, pp. 163–191. Academic Press, London (1985)CrossRefGoogle Scholar
  11. Das, I., Dennis, J.E.: A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Structural Optimization 14(1), 63–69 (1997)CrossRefGoogle Scholar
  12. Deb, K., Chaudhuri, S., Miettinen, K.: Towards estimating nadir objective vector using evolutionary approaches. In: Keijzer, M., et al. (eds.) Proceedings of the 8th Annual Genetic and Evolutionary Computation Conference (GECCO-2006), Seattle, vol. 1, pp. 643–650. ACM Press, New York (2006)Google Scholar
  13. deNeufville, R., McCord, M.: Unreliable measurement of utility: Significant problems for decision analysis. In: Brans, J.P. (ed.) Operational Research ’84, pp. 464–476. Elsevier, Amsterdam (1984)Google Scholar
  14. Edgeworth, F.Y.: Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences. C. Kegan Paul & Co., London (1881), University Microfilms International (Out-of-Print Books on Demand) (1987)MATHGoogle Scholar
  15. Fandel, G.: Group decision making: Methodology and applications. In: Bana e Costa, C. (ed.) Readings in Multiple Criteria Decision Aid, pp. 569–605. Berlin (1990)Google Scholar
  16. Fishburn, P.C.: Lexicographic orders, utilities and decision rules: A survey. Management Science 20(11), 1442–1471 (1974)MathSciNetCrossRefMATHGoogle Scholar
  17. Flavell, R.B.: A new goal programming formulation. Omega 4(6), 731–732 (1976)CrossRefGoogle Scholar
  18. Gass, S., Saaty, T.: The computational algorithm for the parametric objective function. Naval Research Logistics Quarterly 2, 39–45 (1955)MathSciNetCrossRefGoogle Scholar
  19. Geoffrion, A.M.: Proper efficiency and the theory of vector maximization. Journal of Mathematical Analysis and Applications 22(3), 618–630 (1968)MathSciNetCrossRefMATHGoogle Scholar
  20. Haimes, Y.Y., Lasdon, L.S., Wismer, D.A.: On a bicriterion formulation of the problems of integrated system identification and system optimization. IEEE Transactions on Systems, Man, and Cybernetics 1, 296–297 (1971)MathSciNetMATHGoogle Scholar
  21. Hwang, C.-L., Lin, M.-J.: Group Decision Making under Multiple Criteria: Methods and Applications. Springer, New York (1987)CrossRefMATHGoogle Scholar
  22. Hwang, C.L., Masud, A.S.M.: Multiple Objective Decision Making – Methods and Applications: A State-of-the-Art Survey. Springer, Berlin (1979)CrossRefMATHGoogle Scholar
  23. Ignizio, J.P.: Introduction to Linear Goal Programming. Sage Publications, Beverly Hills (1985)CrossRefMATHGoogle Scholar
  24. Jahn, J.: Vector Optimization. Springer, Berlin (2004)CrossRefMATHGoogle Scholar
  25. Jones, D.F., Tamiz, M., Mirrazavi, S.K.: Intelligent solution and analysis of goal programmes: the GPSYS system. Decision Support Systems 23(4), 329–332 (1998)CrossRefGoogle Scholar
  26. Kaliszewski, I.: Quantitative Pareto Analysis by Cone Separation Technique. Kluwer, Dordrecht (1994)CrossRefMATHGoogle Scholar
  27. Keeney, R.L., Raiffa, H.: Decisions with Multiple Objectives: Preferences and Value Tradeoffs. Wiley, Chichester (1976)MATHGoogle Scholar
  28. Koopmans, T.: Analysis and production as an efficient combination of activities. In: Koopmans, T. (ed.) Activity Analysis of Production and Allocation: Proceedings of a Conference, pp. 33–97. Wiley, New York (1951), Yale University Press, London (1971)Google Scholar
  29. Korhonen, P., Salo, S., Steuer, R.E.: A heuristic for estimating nadir criterion values in multiple objective linear programming. Operations Research 45(5), 751–757 (1997)CrossRefMATHGoogle Scholar
  30. Kuhn, H., Tucker, A.: Nonlinear programming. In: Neyman, J. (ed.) Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 481–492. University of California Press, Berkeley (1951)Google Scholar
  31. Lotov, A.V., Bushenkov, V.A., Kamenev, G.K.: Interactive Decision Maps. Approximation and Visualization of Pareto Frontier. Kluwer Academic Publishers, Boston (2004)CrossRefMATHGoogle Scholar
  32. Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989)CrossRefGoogle Scholar
  33. Luque, M., Miettinen, K., Eskelinen, P., Ruiz, F.: Incorporating preference information in interactive reference point methods for multiobjective optimization. Omega 37(2), 450–462 (2009)CrossRefGoogle Scholar
  34. Makarov, E.K., Rachkovski, N.N.: Unified representation of proper efficiency by means of dilating cones. Journal of Optimization Theory and Applications 101(1), 141–165 (1999)MathSciNetCrossRefMATHGoogle Scholar
  35. Marler, R., Arora, J.: Survey of multi-objective optimization methods for engineering. Structural and Multidisciplinary Optimization 26(6), 369–395 (2004)MathSciNetCrossRefMATHGoogle Scholar
  36. Miettinen, K.: Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Boston (1999)MATHGoogle Scholar
  37. Miettinen, K.: Graphical illustration of Pareto optimal solutions. In: Tanino, T., Tanaka, T., Inuiguchi, M. (eds.) Multi-Objective Programming and Goal Programming: Theory and Applications, pp. 197–202. Springer, Berlin (2003)CrossRefGoogle Scholar
  38. Miettinen, K., Mäkelä, M.M., Kaario, K.: Experiments with classification-based scalarizing functions in interactive multiobjective optimization. European Journal of Operational Research 175(2), 931–947 (2006)CrossRefMATHGoogle Scholar
  39. Miettinen, K., Molina, J., González, M., Hernández-Díaz, A., Caballero, R.: Using box indices in supporting comparison in multiobjective optimization. European Journal of Operational Research, to appear (2008), doi:10.1016/j.ejor.2008.05.103Google Scholar
  40. Pareto, V.: Cours d’Economie Politique. Rouge, Lausanne (1896)Google Scholar
  41. Pareto, V.: Manuale di Economia Politica. Piccola Biblioteca Scientifica, Milan (1906), Translated into English by Schwier, A.S., Manual of Political Economy, MacMillan, London (1971)Google Scholar
  42. Podinovski, V.V.: Criteria importance theory. Mathematical Social Sciences 27(3), 237–252 (1994)MathSciNetCrossRefGoogle Scholar
  43. Rodríguez Uría, M., Caballero, R., Ruiz, F., Romero, C.: Meta-goal programming. European Journal of Operational Research 136(2), 422–429 (2002)CrossRefMATHGoogle Scholar
  44. Romero, C.: Handbook of Critical Issues in Goal Programming. Pergamon Press, Oxford (1991)MATHGoogle Scholar
  45. Rommelfanger, H., Slowinski, R.: Fuzzy linear programming with single or multiple objective functions. In: Slowinski, R. (ed.) Fuzzy Sets in Decision Analysis, Operations Research and Statistics, pp. 179–213. Kluwer Academic Publishers, Boston (1998)CrossRefGoogle Scholar
  46. Rosenthal, R.E.: Principles of Multiobjective Optimization. Decision Sciences 16(2), 133–152 (1985)CrossRefGoogle Scholar
  47. Roy, B., Mousseau, V.: A theoretical framework for analysing the notion of relative importance of criteria. Journal of Multi-Criteria Decision Analysis 5(2), 145–159 (1996)CrossRefMATHGoogle Scholar
  48. Ruzika, S., Wiecek, M.M.: Approximation methods in multiobjective programming. Journal of Optimization Theory and Applications 126(3), 473–501 (2005)MathSciNetCrossRefMATHGoogle Scholar
  49. Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization. Academic Press, Orlando (1985)MATHGoogle Scholar
  50. Steuer, R.E.: Multiple Criteria Optimization: Theory, Computation, and Application. Wiley, New York (1986)MATHGoogle Scholar
  51. Tanner, L.: Selecting a text-processing system as a qualitative multiple criteria problem. European Journal of Operational Research 50(2), 179–187 (1991)CrossRefGoogle Scholar
  52. Vincke, P.: Multicriteria Decision-Aid. Wiley, Chichester (1992)MATHGoogle Scholar
  53. Weistroffer, H.R.: Careful usage of pessimistic values is needed in multiple objectives optimization. Operations Research Letters 4(1), 23–25 (1985)CrossRefMATHGoogle Scholar
  54. Wierzbicki, A.P.: A mathematical basis for satisficing decision making. Mathematical Modelling 3, 391–405 (1982)MathSciNetCrossRefMATHGoogle Scholar
  55. Wierzbicki, A.P.: On the completeness and constructiveness of parametric characterizations to vector optimization problems. OR Spectrum 8(2), 73–87 (1986)MathSciNetCrossRefMATHGoogle Scholar
  56. Wierzbicki, A.P.: Reference point approaches. In: Gal, T., Stewart, T.J., Hanne, T. (eds.) Multicriteria Decision Making: Advances in MCDM Models, Algorithms, Theory, and Applications, pp. 9-1–9-39. Kluwer, Boston (1999)Google Scholar
  57. Wierzbicki, A.P.: Reference point methodology. In: Wierzbicki, A.P., Makowski, M., Wessels, J. (eds.) Model-Based Decision Support Methodology with Environmental Applications, pp. 71–89. Kluwer Academic Publishers, Dordrecht (2000)CrossRefGoogle Scholar
  58. Yu, P.L.: A class of solutions for group decision problems. Management Science 19(8), 936–946 (1973)MathSciNetCrossRefMATHGoogle Scholar
  59. Zadeh, L.: Optimality and non-scalar-valued performance criteria. IEEE Transactions on Automatic Control 8, 59–60 (1963)CrossRefGoogle Scholar
  60. Zeleny, M.: Compromise programming. In: Cochrane, J.L., Zeleny, M. (eds.) Multiple Criteria Decision Making, pp. 262–301. University of South Carolina, Columbia, SC (1973)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Kaisa Miettinen
    • 1
  1. 1.Department of Mathematical Information TechnologyUniversity of JyväskyläFinland

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