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Systems of Linear Equations with Fuzzy Set Data: Weak Solvability and Weak Admissibility

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Cybernetics and Systems Analysis Aims and scope

Abstract

The notion of a fuzzy linear system of equations as a set of five special interval systems of equations is introduced. The notions of weak and strong solvability (admissibility) of a fuzzy linear system of equations in five grades (crisp, quasi-crisp, semi-crisp, quasi-fuzzy, and fuzzy) are introduced. The criteria of weak solvability and admissibility of fuzzy linear systems of equations in the five grades are substantiated. Other properties of fuzzy systems and their weak solutions in all the five grades are proved.

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References

  1. L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning,” in: K. S. Fu et al. (eds.), Learning Systems and Intelligent Robots, Plenum Press, New York (1974).

    Google Scholar 

  2. L. A. Zadeh, “Fuzzy sets and their application to pattern recognition and clustering analysis,” in: J. V. Ryzin (ed.), Classification and Clustering, Academic Press, New York–San Francisco–London (1977).

    Google Scholar 

  3. A. Kaufmann, Introduction to the Theory of Fuzzy Subsets, Academic Press (1975).

  4. A. Kaufmann and J. Gil Aluja, Introducción de la teoria de los subconjuntos borrosos a la gestión de las empresas (Introduction of the Fuzzy Set Theory to Enterprise Management), Editorial Milladoiro, Santiago de Compostela (1986).

    Google Scholar 

  5. I. V. Sergienko and M. F. Kaspshitskaya, “Using concepts of fuzzy mathematics for formalization and solution of combinatorial optimization problems,” Cybern. Syst. Analysis, 31, No. 2, 293–296 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  6. I. V. Sergienko, I. N. Parasyuk, and M. F. Kaspshitskaya, “A fuzzy problem of multiparametric choice of optimal solutions,” Cybern. Syst. Analysis, 39, No. 2, 163–173 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  7. I. N. Parasyuk and S. V. Ershov, “Transformations of fuzzy graphs specified by FD-grammars,” Cybern. Syst. Analysis, 43, No. 2, 129–146 (2007).

    Article  MathSciNet  Google Scholar 

  8. D. A. Pospelov (ed.), Fuzzy Sets in Models of Control and Artificial Intelligence [in Russian], Nauka, Moscow (1986).

    Google Scholar 

  9. O. O. Iemets and O. O. Yemets, Solving Combinatorial Optimization Problems on Fuzzy Sets [in Ukrainian], PUET, Poltava (2011), http://dspace.uccu.org.ua/handle/123456789/352.

    Google Scholar 

  10. O. O. Iemets and O. O. Yemets, “Operations and relations on fuzzy numbers,” Nauk. Visti NTUU “KPI,” No. 5, 39–46 (2008).

    Google Scholar 

  11. O. O. Iemets and O. O. Yemets, “Constructing a mathematical model for a combinatorial problem of packing fuzzy-dimensional rectangles,” Nauk. Visti NTUU “KPI,” No. 6, 25–33 (2008).

    Google Scholar 

  12. G. A. Donets and A. O. Yemets, “The statement and solution of the knapsack problem with fuzzy data,” J. Autom. Inform. Sci., 41, No. 9, 1–13 (2009).

    Article  Google Scholar 

  13. T. A. Parfonova and O. A. Yemets, “Operations on fuzzy numbers with bearer of the cardinality of continuum for modeling in combinatorial optimization,” J. Autom. Inform. Sci., 42, No. 4, 19–36 (2010).

    Article  Google Scholar 

  14. O. A. Iemets, A. O. Yemets, and T. A. Parfonova, “Branch and bound method for optimization problems on fuzzy sets,” J. Autom. Inform. Sci., 45, No. 4, 23–29 (2013).

    Article  Google Scholar 

  15. Yu. P. Zaichenko, Operations Research: Fuzzy Optimization. A Handbook [in Russian], Vyshcha Shkola, Kyiv (1991).

    Google Scholar 

  16. O. V. Seraya and L. G. Raskin, “Solving systems of linear algebraic equations with fuzzy parameters,” in: Open Information and Computer Integrated Technologies [in Russian], Issue 31, Kharkov (2006), pp. 233–241.

  17. L. G. Raskin, and O. V. Seraya, Fuzzy Mathematics. Fundamentals of the Theory. Applications [in Russian], Parus, Kharkov (2008).

    Google Scholar 

  18. O. O. Iemets and O. O. Yemets, “Reduction of fuzzy numbers with discrete carrier,” in: Proc. Intern. Sci. Conf. Intellectual systems of decision-making and problems of computational intelligence” (ISDMCI’2012, Yevpatoria, May 27–31, 2012), KhNTU, Kherson (2012), pp. 361–362.

  19. O. O. Iemets and O. O. Yemets’, “About the problem of growing of a discrete fuzzy number carrier during algebraic operations,” in: Proc. 20th Intern. Conf. Problems of Decision Making under Uncertainties: Abstracts (Sept. 17–21, 2012, Brno, Czech Republic), Kyiv (2012), pp. 117–124.

  20. T. Saaty, Decision Making. The Hierarchy Analysis Method [Russian translation], Radio i Svyaz’, Moscow (1993).

    Google Scholar 

  21. M. Fiedler, J. Nedoma, J. Ramik, et al., Linear Optimization Problems with Inexact Data, Springer, New York (2006).

    MATH  Google Scholar 

  22. W. Oettli and W. Prager, “Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides,” Numer. Mathem., No. 6, 405–409 (1964).

    Article  MATH  MathSciNet  Google Scholar 

  23. L. G. Khachiyan, “Polynomial algorithm in linear programming,” Dokl. AN SSSR, 244, 1093–1096 (1979).

    MATH  MathSciNet  Google Scholar 

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Correspondence to I. V. Sergienko.

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Translated from Kibernetika i Sistemnyi Analiz, No. 2, March–April, 2014, pp. 33–43.

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Sergienko, I.V., Iemets, O.O. & Yemets, O.O. Systems of Linear Equations with Fuzzy Set Data: Weak Solvability and Weak Admissibility. Cybern Syst Anal 50, 191–200 (2014). https://doi.org/10.1007/s10559-014-9606-8

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  • DOI: https://doi.org/10.1007/s10559-014-9606-8

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