Multi-Objective and Reference Point Optimization Tools

  • Andrzej P. Wierzbicki
Part of the The International Institute for Applied Systems Analysis book series (MMTA, volume 9)


This chapter presents in more detail the mathematical background, concepts, and tools for multi-objective optimization and reference point methodology, more generally presented in Chapter 4. Section 8.1 presents a comprehensive discussion of various types of Pareto-optimality and efficiency. Additional material on estimating objective ranges and on objective aggregation is discussed in Section 8.2. A detailed presentation of various types of reference points and achievement functions is contained in Section 8.3. Section 8.4 presents further material on weighted and neutral compromise solutions, together with their relations to reference point approaches. Section 8.5 provides brief comments on tools for multi-objective optimization.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 2.
    This point is not sufficiently addressed in many books on multi-objective decision support, although they do often concentrate on the use of importance weights (e.g., Janssen, 1992).Google Scholar
  2. 4.
    If p> 1, then the use of weighting coefficients to control the selection of efficient points might not so easily result in discontinuity, as in the linear case; however, the dependence of efficient points on weighting coefficients does not remain transparent enough. There are cases where changing weighting coefficients produces the reverse effects from those expected (see e.g., Nakayama, 1994).Google Scholar
  3. 5.
    For a more detailed theory of such functions, see e.g., Wierzbicki (1986).Google Scholar
  4. 8.
    The reference point approach was originally developed from investigations of penalty techniques applied to multi-objective optimization (see Wierzbicki, 1980a).Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Andrzej P. Wierzbicki

There are no affiliations available

Personalised recommendations