Soft Computing

, Volume 22, Issue 9, pp 3077–3095 | Cite as

A new attitude coupled with fuzzy thinking for solving fuzzy equations

  • T. Allahviranloo
  • I. Perfilieva
  • F. Abbasi
Methodologies and Application


With the development on the theory of fuzzy numbers, one of the major areas that emerged for the application of these fuzzy numbers is the solution of equations whose parameters are fuzzy numbers. The classical methods, involving the extension principle and \(\alpha \)-cuts, are too restrictive for solving fuzzy equations because very often there is no solution or very strong conditions must be placed on the equations so that there will be a solution. These facts motivated us to solve fuzzy equations with a new attitude. According to the new fuzzy arithmetic operations based on TA (in the domain of the transmission average of support), we discuss a new attitude solving fuzzy equations: \(A+X=B\), \(AX=B\), \(AX+B=C\), \(AX^{2}=B\), \(AX^{2}+B=C\) and \(AX^{2}+BX+C=D.\) Through theoretical analysis, by illustrative examples and computational results, we show that the proposed approach is more general and straightforward.


Fuzzy arithmetics Fuzzy par Ambiguity rank Fuzzy equation Extension principle (EP) Transmission average (TA) 



The authors really appreciate Prof. Didier Dubois for his useful comments to improve the quality of the paper.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants performed by any of the authors.


  1. Abbasi F, Allahviranloo T, Abbasbandy S (2015) A new attitude coupled with fuzzy thinking to fuzzy rings and fields. J Intell Fuzzy Syst 29:851–861MathSciNetCrossRefzbMATHGoogle Scholar
  2. Biacino L, Lettieri A (1989) Equation with fuzzy numbers. Inf Sci 47(1):63–76MathSciNetCrossRefGoogle Scholar
  3. Buckley JJ (1992) Solving fuzzy equations. Fuzzy Sets Syst 50(1):1–14MathSciNetCrossRefzbMATHGoogle Scholar
  4. Buckley JJ, Qu Y (1990) Solving linear and quadratic equations. Fuzzy Sets Syst 38(1):48–59MathSciNetCrossRefzbMATHGoogle Scholar
  5. Buckley JJ et al (1997) Solving fuzzy equations using neural nets. Fuzzy Sets Syst 86:271–278CrossRefzbMATHGoogle Scholar
  6. Dubois D, Prade H (1980) Fuzzy sets and systems: theory and applications. Academic Press, New YorkGoogle Scholar
  7. Dubois D, Prade H (1984) Fuzzy set theoretic differences and inclusions and their use in the analysis of fuzzy equations. Control Cybern (Warshaw) 13:129–146MathSciNetzbMATHGoogle Scholar
  8. Fullór R (1998) Fuzzy reasoning and fuzzy optimization. On Leave from Department of Operations Research, Eötvös Lorand University, BudapestGoogle Scholar
  9. Jain R (1976a) Outline of an approach for the analysis of fuzzy systems. Int J Control 23(5):627–640MathSciNetCrossRefzbMATHGoogle Scholar
  10. Jain R (1976b) Tolerance analysis using fuzzy sets. Int J Syst Sci 7(12):1393–1401CrossRefzbMATHGoogle Scholar
  11. Jain R (1977) A procedure for multiple aspect decision making using fuzzy sets. Int J Syst Sci 8(1):1–7MathSciNetCrossRefzbMATHGoogle Scholar
  12. Jiang H (1986) The approach to solving simultaneous linear equations that coefficients are fuzzy numbers. J Natl Univ Def Technol (Chin) 3:96–102Google Scholar
  13. Kawaguchi MF, Date T (1993) A calculation method for solving fuzzy arithmetic equation with triangular norms. In: Proceedings of 2nd IEEE international conference on fuzzy systems (FUZZY-IEEE), San Francisco, pp 470–476Google Scholar
  14. Klir GJ, Yuan B (1995) Fuzzy sets and fuzzy logic: theory and applications. Prentice-Hall PTR, Upper Saddlie RiverzbMATHGoogle Scholar
  15. Mazarbhuiya FA et al (2011) Solution of the fuzzy equation A \(+\) X \(=\) B using the method of superimposition. Appl Math 2:1039–1045MathSciNetCrossRefGoogle Scholar
  16. Mizumoto M, Tanaka K (1982) Algebraic properties of fuzzy numbers. In: Gupta MM, Ragade RK, Yager RR (eds) Advances in fuzzy set theory and applications. North-Holland, AmsterdamGoogle Scholar
  17. Sanchez E (1977) Solutions in composite fuzzy relation equation: application to medical diagnosis in Brouwerian logic. In: Gupta M, Saridis GN, Gaines BR (eds) Fuzzy automata and decision processes. North-Holland, New York, pp 221–234Google Scholar
  18. Sanchez E (1984) Solution of fuzzy equations with extend operations. Fuzzy Sets Syst 12:273–248CrossRefGoogle Scholar
  19. Stefanini L (2010) A generalization of hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets Syst 161:1564–1584MathSciNetCrossRefzbMATHGoogle Scholar
  20. Wang X, Ha M (1994) Solving a system of fuzzy linear equations. In: Delgado M, Kacpryzykand J, Verdegay JL, Vila A (eds) Fuzzy optimisation: recent advances. Physica-Verlag, Heildelberg, pp 102–108Google Scholar
  21. Wasowski J (1997) On solutions to fuzzy equations. Control Cybern 26:653–658MathSciNetzbMATHGoogle Scholar
  22. Yager RR (1977) Building fuzzy systems models, vol 5, Nato conference series, applied general systems research: recent developments and trends. Plenum Press, New York, pp 313–320Google Scholar
  23. Yager RR (1980) On the lack of inverses in fuzzy arithmetic. Fuzzy Sets Syst 4:73–82MathSciNetCrossRefzbMATHGoogle Scholar
  24. Zhao R, Govind R (1991) Solutions of algebraic equations involving generalised fuzzy number. Inf Sci 56:199–243CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Science and Research BranchIslamic Azad UniversityTehranIran
  2. 2.Institute for Research and Applications of Fuzzy ModelingUniversity of OstravaOstravaCzech Republic
  3. 3.Department of Mathematics, Ayatollah Amoli BranchIslamic Azad UniversityAmolIran

Personalised recommendations