Abstract
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.
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Abbasi F, Allahviranloo T, Abbasbandy S (2015) A new attitude coupled with fuzzy thinking to fuzzy rings and fields. J Intell Fuzzy Syst 29:851–861
Biacino L, Lettieri A (1989) Equation with fuzzy numbers. Inf Sci 47(1):63–76
Buckley JJ (1992) Solving fuzzy equations. Fuzzy Sets Syst 50(1):1–14
Buckley JJ, Qu Y (1990) Solving linear and quadratic equations. Fuzzy Sets Syst 38(1):48–59
Buckley JJ et al (1997) Solving fuzzy equations using neural nets. Fuzzy Sets Syst 86:271–278
Dubois D, Prade H (1980) Fuzzy sets and systems: theory and applications. Academic Press, New York
Dubois D, Prade H (1984) Fuzzy set theoretic differences and inclusions and their use in the analysis of fuzzy equations. Control Cybern (Warshaw) 13:129–146
Fullór R (1998) Fuzzy reasoning and fuzzy optimization. On Leave from Department of Operations Research, Eötvös Lorand University, Budapest
Jain R (1976a) Outline of an approach for the analysis of fuzzy systems. Int J Control 23(5):627–640
Jain R (1976b) Tolerance analysis using fuzzy sets. Int J Syst Sci 7(12):1393–1401
Jain R (1977) A procedure for multiple aspect decision making using fuzzy sets. Int J Syst Sci 8(1):1–7
Jiang H (1986) The approach to solving simultaneous linear equations that coefficients are fuzzy numbers. J Natl Univ Def Technol (Chin) 3:96–102
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 470–476
Klir GJ, Yuan B (1995) Fuzzy sets and fuzzy logic: theory and applications. Prentice-Hall PTR, Upper Saddlie River
Mazarbhuiya FA et al (2011) Solution of the fuzzy equation A \(+\) X \(=\) B using the method of superimposition. Appl Math 2:1039–1045
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, Amsterdam
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 221–234
Sanchez E (1984) Solution of fuzzy equations with extend operations. Fuzzy Sets Syst 12:273–248
Stefanini L (2010) A generalization of hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets Syst 161:1564–1584
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 102–108
Wasowski J (1997) On solutions to fuzzy equations. Control Cybern 26:653–658
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 313–320
Yager RR (1980) On the lack of inverses in fuzzy arithmetic. Fuzzy Sets Syst 4:73–82
Zhao R, Govind R (1991) Solutions of algebraic equations involving generalised fuzzy number. Inf Sci 56:199–243
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The authors really appreciate Prof. Didier Dubois for his useful comments to improve the quality of the paper.
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Allahviranloo, T., Perfilieva, I. & Abbasi, F. A new attitude coupled with fuzzy thinking for solving fuzzy equations. Soft Comput 22, 3077–3095 (2018). https://doi.org/10.1007/s00500-017-2562-2
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DOI: https://doi.org/10.1007/s00500-017-2562-2