Fuzzy Optimization and Decision Making

, Volume 15, Issue 1, pp 21–31 | Cite as

Generalized bipolar product and sum

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Abstract

Aggregation functions on \([0,1]\) with annihilator 0 can be seen as a generalized product on \([0,1]\). We study the generalized product on the bipolar scale \([-1,1]\), stressing the axiomatic point of view ( compare also Greco et al. in IPMU 2014, CCIS 444, Springer International Publishing, New York, 2014). Based on newly introduced bipolar properties, such as the bipolar monotonicity, bipolar unit element, bipolar idempotent element, several kinds of generalized bipolar product are introduced and studied. A special stress is put on bipolar semicopulas, bipolar quasi-copulas and bipolar copulas. Inspired by the truncated sum on \([-1,1]\) we introduce also the class of generalized bipolar sums, which differ from uninorms due to the non-associativity.

Keywords

Aggregation function Bipolar copula Bipolar scale Bipolar semicopula Symmetric minimum 
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Notes

Acknowledgments

The work of R. Mesiar on this paper was supported by the European Regional Development Fund in the IT4Innovations Centre of Excellence Project Reg. No. CZ.1.05/1.1.00/ 02.0070, and the grant VEGA 1/0420/15.

References

  1. Bassan, B., & Spizzichino, F. (2005). Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes. Journal of Multivariate Analysis, 93(2), 313–339.CrossRefMathSciNetMATHGoogle Scholar
  2. Beliakov, G., Pradera, A., & Calvo, T. (2008). Aggregation functions: A guide for practitioners. Berlin: Studies in Fuziness and Soft Computing, Springer.Google Scholar
  3. Calvo, T., De Baets, B., & Fodor, J. (2001). The functional equations of Frank and Alsina for uninorms and nullnorms. Fuzzy Sets and Systems, 120(3), 385–394.CrossRefMathSciNetMATHGoogle Scholar
  4. Dubois, D., & Prade, H. (2008). An introduction to bipolar representations of information and preference. International Journal of Intelligent Systems, 23(8), 866–877.CrossRefMATHGoogle Scholar
  5. Durante, F., Mesiar, R., & Papini, P. L. (2008). The lattice-theoretic structure of the sets of triangular norms and semi-copulas. Nonlinear Analysis: Theory, Methods & Applications, 69(1), 46–52.CrossRefMathSciNetMATHGoogle Scholar
  6. Durante, F., & Sempi, C. (2005). Semicopulæ. Kybernetika, 41(3), 315–328.MathSciNetMATHGoogle Scholar
  7. Grabisch, M. (2003). The symmetric Sugeno integral. Fuzzy Sets and Systems, 139(3), 473–490.CrossRefMathSciNetMATHGoogle Scholar
  8. Grabisch, M., Greco, S., & Pirlot, M. (2008). Bipolar and bivariate models in multicriteria decision analysis: Descriptive and constructive approaches. International Journal of Intelligent Systems, 23(9), 930–969.CrossRefMATHGoogle Scholar
  9. Grabisch, M., Marichal, J. L., Mesiar, R., & Pap, E. (2009). Aggregation functions (Encyclopedia of mathematics and its applications). Cambridge University PressGoogle Scholar
  10. Greco, S., Mesiar, R., & Rindone, F. (2012). The bipolar universal integral. In IPMU 2012, Part III, CCIS 300 (pp. 360–369).Google Scholar
  11. Greco, S., Mesiar, R., & Rindone, F. (2015). Bipolar semicopulas. Fuzzy Sets and Systems, 268, 141–148.CrossRefMathSciNetGoogle Scholar
  12. Greco, S., Mesiar, R., & Rindone, F. (2014). Generalized product. In IPMU 2014, CCIS 444 (pp. 289–295). Springer International Publishing.Google Scholar
  13. Greco, S., & Rindone, F. (2013). Bipolar fuzzy integrals. Fuzzy Sets and Systems, 220, 21–33.CrossRefMathSciNetMATHGoogle Scholar
  14. Greco, S., & Rindone, F. (2014). The bipolar Choquet integral representation. Theory and Decision, 77(1), 1–29.CrossRefMathSciNetMATHGoogle Scholar
  15. Hájek, P. (1998). Metamathematics of fuzzy logic (Vol. 4). Berlin: Springer.MATHGoogle Scholar
  16. Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica: Journal of the Econometric Society, 47(2), 263–291.CrossRefMATHGoogle Scholar
  17. Klement, E. P., Mesiar, E., & Pap, R. (2000). Triangular norms. Dordrecht: Kluwer.CrossRefMATHGoogle Scholar
  18. Klement, E. P., Mesiar, R., & Pap, E. (2010). A universal integral as common frame for Choquet and Sugeno integral. IEEE Transactions on Fuzzy Systems, 18(1), 178–187.CrossRefGoogle Scholar
  19. Mesiarová-Zemánková, A. (2015). Multi-polar t-conorms and uninorms. Information Sciences, 301, 227–240.Google Scholar
  20. Mesiarová-Zemánková, A., & Ahmad, K. (2014). Extended multi-polarity and multi-polar-valued fuzzy sets. Fuzzy Sets and Systems, 234, 61–78.CrossRefMathSciNetGoogle Scholar
  21. Mesiarová-Zemánková, A., & Ahmad, K. (2013). Multi-polar Choquet integral. Fuzzy Sets and Systems, 220, 1–20.CrossRefMathSciNetMATHGoogle Scholar
  22. Nelsen, R. B. (2006). An introduction to copulas. New York: Springer.MATHGoogle Scholar
  23. Nelsen, R. B., & Flores, M. Ú. (2005). The lattice-theoretic structure of sets of bivariate copulas and quasi-copulas. Comptes Rendus Mathematique, 341(9), 583–586.CrossRefMathSciNetMATHGoogle Scholar
  24. Schweizer, B., & Sklar, A. (1960). Statistical metric spaces. Pacific Journal of Mathematics, 10, 313–334.CrossRefMathSciNetMATHGoogle Scholar
  25. Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5(4), 297–323.CrossRefMATHGoogle Scholar
  26. Wang, Z., & Klir, G. J. (2009). Generalized measure theory (Vol. 25). NewYork: Springer.CrossRefMATHGoogle Scholar
  27. Weber, S. (1984). \(\perp \)-decomposable measures and integrals for archimedean t-conorms \(\perp \). Journal of Mathematical Analysis and Applications, 101(1), 114–138.CrossRefMathSciNetMATHGoogle Scholar
  28. Yager, R., & Rybalov, A. (2011). Bipolar aggregation using the uninorms. Fuzzy Optimization and Decision Making, 10(1), 59–70.CrossRefMathSciNetMATHGoogle Scholar
  29. Yager, R., & Rybalov, A. (1996). Uninorm aggregation operators. Fuzzy Sets and Systems, 80(1), 111–120.CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Economics and BusinessCataniaItaly
  2. 2.Centre of Operations Research and Logistics (CORL), Portsmouth Business SchoolUniversity of PortsmouthPortsmouthUK
  3. 3.Department of Mathematics and Descriptive Geometry, Faculty of Civil EngineeringSlovak University of TechnologyBratislavaSlovakia

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