Strongly Proper Efficient Solutions: Efficient Solutions with Bounded Trade-Offs

  • Kazhal Khaledian
  • Esmaile Khorram
  • Majid Soleimani-damaneh


In multiple-objective optimization literature, a properly efficient solution has been interpreted as a point in which the trade-offs between all objectives are bounded. In this paper, it is shown that this boundedness does not necessarily hold for problems with three or more objective functions. It is possible that in a properly efficient solution the trade-offs between some objectives are unbounded. To overcome this, in this paper strongly proper efficient solutions are introduced, in which the trade-offs between all objectives are bounded. This notion is defined in different senses, and the relationships between them are investigated. In addition to theoretical discussions, some clarifying examples are given.


Multiple-objective optimization Proper efficiency  Strong proper efficiency Trade-off 

Mathematics Subject Classifications

90C29 65K05 



The authors would like to express their gratitude to anonymous referees and the editor of JOTA for their time and effort about the paper. The research of the third author has been supported by a grant from IPM (No. 94260124).


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Kazhal Khaledian
    • 1
  • Esmaile Khorram
    • 1
  • Majid Soleimani-damaneh
    • 2
    • 3
  1. 1.Faculty of Mathematics and Computer SciencesAmirkabir University of TechnologyTehranIran
  2. 2.School of Mathematics, Statistics and Computer Science, College of ScienceUniversity of TehranTehranIran
  3. 3.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran

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