Solution of the Fuzzy Equation *A + X = B* Using the Method of Superimposition

ABSTRACT

Fuzzy equations were solved by using different standard methods. One of the well-known methods is the method of α-cut. The method of superimposition of sets has been used to define arithmetic operations of fuzzy numbers. In this article, it has been shown that the fuzzy equation , where A, X, B are fuzzy numbers can be solved by using the method of superimposition of sets. It has also been shown that the method gives same result as the method of α-cut.

Fuzzy equations were solved by using different standard methods. One of the well-known methods is the method of α-cut. The method of superimposition of sets has been used to define arithmetic operations of fuzzy numbers. In this article, it has been shown that the fuzzy equation , where A, X, B are fuzzy numbers can be solved by using the method of superimposition of sets. It has also been shown that the method gives same result as the method of α-cut.

KEYWORDS

Fuzzy Number, Possibility Distribution, Probability Distribution, Survival Function, Superimposition of Sets, Superimposition of Intervals, α-Cut Method

Fuzzy Number, Possibility Distribution, Probability Distribution, Survival Function, Superimposition of Sets, Superimposition of Intervals, α-Cut Method

Cite this paper

nullF. Mazarbhuiya, A. Mahanta and H. Baruah, "Solution of the Fuzzy Equation*A + X = B* Using the Method of Superimposition," *Applied Mathematics*, Vol. 2 No. 8, 2011, pp. 1039-1045. doi: 10.4236/am.2011.28144.

nullF. Mazarbhuiya, A. Mahanta and H. Baruah, "Solution of the Fuzzy Equation

References

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[2] E. Sanchez, “Solution of Fuzzy Equations with Extended Operations,” Fuzzy Sets and Systems, Vol. 12, 1984, pp. 273-248. doi:10.1016/0165-0114(84)90071-X

[3] J. J. Buckley, “Solving Fuzzy Equations,” Fuzzy Sets and Systems, Vol. 50, No. 1, 1992, pp. 1-14. doi:10.1016/0165-0114(92)90199-E

[4] J. Wasowski, “On Solutions to Fuzzy Equations,” Control and Cybern, Vol. 26, 1997, pp. 653-658.

[5] L. Biacino and A. Lettieri, “Equation with Fuzzy Numbers,” Information Sciences, Vol. 47, No. 1, 1989, pp. 63-76.

[6] H. Jiang, “The Approach to Solving Simultaneous Linear Equations That Coefficients Are Fuzzy Numbers,” Journal of National University of Defence Technology (Chinese), Vol. 3, 1986, pp. 96-102.

[7] J. J. Buckley and Y. Qu, “Solving Linear and Quadratic Equations,” Fuzzy Sets and Systems, Vol. 38, No.1, 1990, pp. 48-59. doi:10.1016/0165-0114(90)90099-R

[8] M. F. Kawaguchi and T. Da-Te, “A Calculation Method for Solving Fuzzy Arithmetic Equation with Triangular Norms,” Proceedings of 2nd IEEE International Conference on Fuzzy Systems (FUZZY-IEEE), San Francisco, 1993, pp. 470-476.

[9] R. Zhao and R. Govind, “Solutions of Algebraic Equations Involving Generalised Fuzzy Number,” Information Sciences, Vol. 56, 1991, pp. 199-243. doi:10.1016/0020-0255(91)90031-O

[10] X. Wang and M. Ha, “Solving a System of Fuzzy Linear Equations,” In: M. Delgado, J. Kacpryzyk, J. L. Verdegay and A. Vila, Eds., Fuzzy Optimisation: Recent Advances, Physica-Verlag, Heildelberg, 1994, pp. 102-108.

[11] G. J. Klir and B. Yuan, “Fuzzy Sets and Fuzzy Logic Theory and Applications,” Prentice Hall of India Pvt. Ltd., 2002.

[12] F. A. Mazarrbhuiya, A. K. Mahanta and H. K. Baruah, “Fuzzy Arithmetic without Using the Method of ?-Cuts,” Bulletin of Pure and Applied Sciences, Vol. 22 E, No. 1, 2003, pp. 45-54.

[13] H. K. Baruah, “Set Superimposition and Its Application to the Theory of Fuzzy sets,” Journal of Assam Science Society, Vol. 10, No. 1-2, 1999, pp. 25-31.

[14] G. Q. Chen, S. C .Lee and S. H. Yu Eden, “Application of Fuzzy Set Theory to Economics,” In: P. P. Wang, Ed., Advances in Fuzzy Sets, Possibility Theory, and Applications, 1983, pp. 277-305.

[15] D. Dubois and H. Prade, “Ranking Fuzzy Numbers in the Setting of Possibility Theory,” Information Science, Vol. 30, No. 3, 1983, pp. 183-224. doi:10.1016/0020-0255(83)90025-7

[16] M. Loeve, “Probability Theory,” Springer Verlag, New York, 1977.

[1] D. Dubois and H. Prade, “Fuzzy Set Theoretic Differences and Inclusions and Their Use in The analysis of Fuzzy Equations,” Control Cybern (Warshaw), Vol. 13, 1984, pp. 129-146.

[2] E. Sanchez, “Solution of Fuzzy Equations with Extended Operations,” Fuzzy Sets and Systems, Vol. 12, 1984, pp. 273-248. doi:10.1016/0165-0114(84)90071-X

[3] J. J. Buckley, “Solving Fuzzy Equations,” Fuzzy Sets and Systems, Vol. 50, No. 1, 1992, pp. 1-14. doi:10.1016/0165-0114(92)90199-E

[4] J. Wasowski, “On Solutions to Fuzzy Equations,” Control and Cybern, Vol. 26, 1997, pp. 653-658.

[5] L. Biacino and A. Lettieri, “Equation with Fuzzy Numbers,” Information Sciences, Vol. 47, No. 1, 1989, pp. 63-76.

[6] H. Jiang, “The Approach to Solving Simultaneous Linear Equations That Coefficients Are Fuzzy Numbers,” Journal of National University of Defence Technology (Chinese), Vol. 3, 1986, pp. 96-102.

[7] J. J. Buckley and Y. Qu, “Solving Linear and Quadratic Equations,” Fuzzy Sets and Systems, Vol. 38, No.1, 1990, pp. 48-59. doi:10.1016/0165-0114(90)90099-R

[8] M. F. Kawaguchi and T. Da-Te, “A Calculation Method for Solving Fuzzy Arithmetic Equation with Triangular Norms,” Proceedings of 2nd IEEE International Conference on Fuzzy Systems (FUZZY-IEEE), San Francisco, 1993, pp. 470-476.

[9] R. Zhao and R. Govind, “Solutions of Algebraic Equations Involving Generalised Fuzzy Number,” Information Sciences, Vol. 56, 1991, pp. 199-243. doi:10.1016/0020-0255(91)90031-O

[10] X. Wang and M. Ha, “Solving a System of Fuzzy Linear Equations,” In: M. Delgado, J. Kacpryzyk, J. L. Verdegay and A. Vila, Eds., Fuzzy Optimisation: Recent Advances, Physica-Verlag, Heildelberg, 1994, pp. 102-108.

[11] G. J. Klir and B. Yuan, “Fuzzy Sets and Fuzzy Logic Theory and Applications,” Prentice Hall of India Pvt. Ltd., 2002.

[12] F. A. Mazarrbhuiya, A. K. Mahanta and H. K. Baruah, “Fuzzy Arithmetic without Using the Method of ?-Cuts,” Bulletin of Pure and Applied Sciences, Vol. 22 E, No. 1, 2003, pp. 45-54.

[13] H. K. Baruah, “Set Superimposition and Its Application to the Theory of Fuzzy sets,” Journal of Assam Science Society, Vol. 10, No. 1-2, 1999, pp. 25-31.

[14] G. Q. Chen, S. C .Lee and S. H. Yu Eden, “Application of Fuzzy Set Theory to Economics,” In: P. P. Wang, Ed., Advances in Fuzzy Sets, Possibility Theory, and Applications, 1983, pp. 277-305.

[15] D. Dubois and H. Prade, “Ranking Fuzzy Numbers in the Setting of Possibility Theory,” Information Science, Vol. 30, No. 3, 1983, pp. 183-224. doi:10.1016/0020-0255(83)90025-7

[16] M. Loeve, “Probability Theory,” Springer Verlag, New York, 1977.