OR Spectrum

, Volume 34, Issue 4, pp 803–816 | Cite as

Trade-off analysis approach for interactive nonlinear multiobjective optimization

Regular Article


When solving multiobjective optimization problems, there is typically a decision maker (DM) who is responsible for determining the most preferred Pareto optimal solution based on his preferences. To gain confidence that the decisions to be made are the right ones for the DM, it is important to understand the trade-offs related to different Pareto optimal solutions. We first propose a trade-off analysis approach that can be connected to various multiobjective optimization methods utilizing a certain type of scalarization to produce Pareto optimal solutions. With this approach, the DM can conveniently learn about local trade-offs between the conflicting objectives and judge whether they are acceptable. The approach is based on an idea where the DM is able to make small changes in the components of a selected Pareto optimal objective vector. The resulting vector is treated as a reference point which is then projected to the tangent hyperplane of the Pareto optimal set located at the Pareto optimal solution selected. The obtained approximate Pareto optimal solutions can be used to study trade-off information. The approach is especially useful when trade-off analysis must be carried out without increasing computation workload. We demonstrate the usage of the approach through an academic example problem.


Multicriteria optimization Interactive methods Trade-off rate Reference point Pareto optimality Decision support 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Chankong V, Haimes YY (1977) The interactive surrogate worth trade-off (ISWT) method for multiobjective decision making. In: Zionts S (eds) Multiple criteria problem solving, vol 155. Springer, Berlin, pp 42–67Google Scholar
  2. Chankong V, Haimes YY (1983) Multiobjective decision making theory and methodology. Elsevier Science Publishing Co., Inc., New YorkGoogle Scholar
  3. Kuk H, Tanino T, Tanaka M (1997) Trade-off analysis for vector optimization problems via scalarization. J Inf Optim Sci 18: 75–87Google Scholar
  4. Miettinen K (1999) Nonlinear multiobjective optimization. Kluwer, BostonGoogle Scholar
  5. Miettinen K (2006) IND-NIMBUS for demanding interactive multiobjective optimization. In: Trzaskalik T (eds) Multiple criteria decision making ’05. The Karol Adamiecki University of Economics in Katowice, Katowice, pp 137–150Google Scholar
  6. Miettinen K, Mäkelä MM (2006) Synchronous approach in interactive multiobjective optimization. Eur J Oper Res 170(3): 909–922CrossRefGoogle Scholar
  7. Miettinen K, Ruiz F, Wierzbicki AP (2008) Introduction to multiobjective optimization: interactive approaches. In: Branke J, Deb K, Miettinen K, Slowinski R (eds) Multiobjective optimization: interactive and evolutionary approaches. Springer, Berlin, pp 27–57Google Scholar
  8. Nakayama H, Sawaragi Y (1984) Satisficing trade-off method for multiobjective programming. In: Grauer M, Wierzbicki AP (eds) Interactive decision analysis. Springer, New York, pp 113–122Google Scholar
  9. Nocedal J, Wright SJ (2006) Numerical Optimization, 2nd edn. Springer, New YorkGoogle Scholar
  10. Sakawa M (1982) Interactive multiobjective decision making by the sequential proxy optimization technique: SPOT. Eur J Oper Res 9: 386–396CrossRefGoogle Scholar
  11. Sakawa M, Yano H (1990) Trade-off rates in the hyperplane method for multiobjective optimization problems. Eur J Oper Res 44: 105–118CrossRefGoogle Scholar
  12. Sindhya K, Deb K, Miettinen K (2011) Improving convergence of evolutionary multi-objective optimization with local search: a concurrent hybrid algorithm. Nat Comput. doi: 10.1007/s11047-011-9250-4
  13. Tappeta RV, Renaud JE (1999) Interactive multiobjective optimization procedure. AIAA J 37: 881–889CrossRefGoogle Scholar
  14. Wierzbicki AP (1986) On completeness and constructiveness of parametric characterizations to vector optimization problems. OR Spektrum 8: 73–87CrossRefGoogle Scholar
  15. Yang J-B (1999) Gradient projection and local region search for multiobjective optimisation. Eur J Oper Res 112: 432–459CrossRefGoogle Scholar
  16. Yang J-B, Li D (2002) Normal vector identification and interactive tradeoff analysis using minimax formulation in multiobjective optimization. IEEE Trans Syst Man Cybern Part A: Syst Hum 32(3): 305–319CrossRefGoogle Scholar
  17. Yano H, Sakawa M (1987) Trade-off rates in the weighted Tchebycheff norm method. Large Scale Syst 13: 167–177Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of Mathematical Information TechnologyUniversity of JyväskyläFinland

Personalised recommendations