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Searching for Pareto-Optimal Solutions

  • Igor Kovalenko
  • Yevhen DavydenkoEmail author
  • Alyona Shved
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1080)

Abstract

The problem of narrowing the Pareto set is considered. The existing approaches aimed at solving the problem of finding a set of Pareto optimal solutions has been analyzed. One approach for solving multi-objective problems based on complex using methods for the construction of the Pareto optimal set and evidence theory has been proposed in this paper. The proposed technique allows to evaluate the obtained set of non-dominated alternatives by methods of the evidence theory to find the best (optimal) solution. The proposed approach allows us to obtain a more formalized procedure for narrowing the Pareto set to obtaining a single optimal solution (a single-element Pareto set). The use of the mathematical apparatus of the evidence theory makes it possible to model uncertainty in expert or decision makers judgments (the strict requirement of the “unambiguous” preference of one alternative over the other is removed). Using the proportional conflict redistribution rules for aggregating group expert assessments makes it possible to process expert evidence generated under conflicting, contradiction expert information. Numerical examples of the proposed methodology for integrated application of evidence theory and methods for Pareto set construction to find optimal solutions are given. The results obtained make it possible to improve the quality and effectiveness of finding optimal solutions.

Keywords

Multi-objective problem Pareto set Evidence theory Optimal solution 

Notes

Acknowledgment

This research was partially supported by the state research project “Development of information and communication decision support technologies for strategic decision-making with multiple criteria and uncertainty for military-civilian use” (research project no. 0117U007144, financed by the Government of Ukraine).

References

  1. 1.
    Nogin, V.D.: Priniatie reshenii v mnogokriterialnoi srede: kolichestvennyi podkhod, 2nd edn. Fizmatlit, Moscow (2005). (in Russian)Google Scholar
  2. 2.
    Bogdanova, A.V., Nogin, V.D.: Reduction of the Pareto set based on some compound information on the relative importance of criteria. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, vol. 2, pp. 1–16 (2007). (in Russian)Google Scholar
  3. 3.
    Nogin, V.D., Volkova, N.A.: Evolution of the Edgeworth-Pareto principle. Tavricheskiĭ Vestnik Informatiki i Matematiki 1, 21–33 (2006). (in Russian)zbMATHGoogle Scholar
  4. 4.
    Zakharov, A.O.: Suzhenie mnozhestva Pareto na osnove vzaimozavisimoi informatsii zamknutogo tipa. Iskusstvennyi Intellect i Priniatie Reshenii 1, 67–81 (2011). (in Russian)Google Scholar
  5. 5.
    Nogin, V.D.: Problema suzheniia mnozhestva Pareto: podkhody k resheniiu. Iskusstvennyi Intellect i Priniatie Reshenii 1, 98–112 (2008). (in Russian)Google Scholar
  6. 6.
    Nogin, V.D.: Algoritm suzheniia mnozhestva Pareto na osnove proizvolnogo konechnogo nabora “kvantov” informatsii. Iskusstvennyi Intellect i Priniatie Reshenii 1, 63–69 (2013). (in Russian)Google Scholar
  7. 7.
    Nogin, V.D.: Suzhenie mnozhestva Pareto na osnove nechetkoi informatsii. Int. J. Inf. Technol. Knowl. 6(2), 157–168 (2012). (in Russian)Google Scholar
  8. 8.
    Podinovskii, V.V.: Vvedenie v teoriiu vazhnosti kriteriev v mnogokriterialnykh zadachakh priniatiia reshenii. Fizmatlit, Moscow (2007). (in Russian)Google Scholar
  9. 9.
    Shafer, G.A.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)zbMATHGoogle Scholar
  10. 10.
    Beynon, M.J., Curry, B., Morgan, P.: The Dempster-Shafer theory of evidence: an alternative approach to multicriteria decision modeling. Omega 28(1), 37–50 (2000)CrossRefGoogle Scholar
  11. 11.
    Dempster, A.P.: A generalization of Bayesian inference. J. Roy. Stat. Soc. B(30), 205–247 (1968)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Nogin, V.D.: Priniatie reshenii pri mnogikh kriteriiakh. IUTAS, SPb (2007). (in Russian)Google Scholar
  13. 13.
    Sentz, K., Ferson, S.: Combination of evidence in Dempster–Shafer theory. Sandia National Laboratories, Albuquerque, New Mexico (2002)Google Scholar
  14. 14.
    Shved, A., Davydenko, Ye.: The analysis of uncertainty measures with various types of evidence. In: 1st International Conference on Data Stream Mining & Processing, Lviv, Ukraine, pp. 61–64. IEEE (2016).  https://doi.org/10.1109/dsmp.2016.7583508
  15. 15.
    Uzga-Rebrovs, O.: Upravlenie neopredelennostiami, vol. 3. Izdevnieciba, Rezekne (2010). (in Russian)Google Scholar
  16. 16.
    Zadeh, L.A.: Review of Shafer’s “A mathematical theory of evidence”. AI Mag. 5(3), 81–83 (1984)Google Scholar
  17. 17.
    Yager, R.R.: On the Dempster-Shafer framework and new combination rules. Inf. Sci. 41(2), 93–137 (1987)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Inagaki, T.: Interdepence between safety-control policy and multiple-sensor schemes via Dempster-Shafer theory. Trans. Reliab. 40(2), 182–188 (1991)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zhang, L.: Representation, independence and combination of evidence in the Dempster-Shafer theory of evidence. In: Yager, R.R., Kacprzyk, J., Fedrizzi, M. (eds.) Advances in the Dempster-Shafer Theory of Evidence, pp. 51–69. Wiley, New York (1994)Google Scholar
  20. 20.
    Smets, Ph.: The combination of evidence in the transferable belief model. Pattern Anal. Mach. Intell. 12, 447–458 (1990)Google Scholar
  21. 21.
    Smarandache, F., Dezert, J.: Advances and Applications of DSmT for Information Fusion, vol. 1. American Research Press, Rehoboth (2004)zbMATHGoogle Scholar
  22. 22.
    Smarandache, F., Dezert, J.: Advances and Applications of DSmT for Information Fusion, vol. 2. American Research Press, Rehoboth (2006)zbMATHGoogle Scholar
  23. 23.
    Kovalenko, I., Davydenko, Ye., Shved, A.: Development of the procedure for integrated application of scenario prediction methods. Eastern-Eur. J. Enterp. Technol. 2/4(98), 31–37 (2019).  https://doi.org/10.15587/1729-4061.2019.163871
  24. 24.
    Beynon, M.J.: DS/AHP method: a mathematical analysis, including an understanding of uncertainty. Eur. J. Oper. Res. 140, 148–164 (2002)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Petro Mohyla Black Sea National UniversityMykolaivUkraine

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