Solving Fuzzy Equations And Fuzzy Linear Systems Based On Fuzzy Structured Element Method

Fuzzy Mathematics is widely applied to engineering technology, scientific management and financial engineering and so on, and some practical problems can be described by fuzzy equations and fuzzy linear systems. However, solving fuzzy equations and fuzzy linear systems which is very difficult, so it is the most important problem and at the same time the most difficult problem in long-term. Solving fuzzy equations and fuzzy linear systems has a very important significance for theoretical and practical application.In view of the difficulties of fuzzy operation and the analytical expression of membership function in traditional solving fuzzy equations and fuzzy linear systems, this paper introduces the fuzzy structured element theory, and puts forward the solving method of fuzzy linear equation, power fuzzy equation, quadratic fuzzy equation and several kinds of fuzzy linear systems, and then systematically establishes analysis techniques of fuzzy structured element of solving fuzzy equations and fuzzy linear systems.Firstly, this paper puts forward the equality constraint operator by defining the operation of two monotone functions based on the method of fuzzy structured element, which improves the content of the equality constraint arithmetic and solves the irreversible problem of multiplication when existing a negative fuzzy number. And the equality constraint operator is applied to fuzzy linear equation, the paper gives the expression of structured element for a fuzzy solution and the necessary and sufficient conditions for a fuzzy solution existence. What's more, a general dual fuzzy linear equation is studied. Moreover, the fuzzy linear equation with rectangular complex fuzzy number and circular wedge complex fuzzy number is also investigated.Secondly, in this paper, power fuzzy number and power fuzzy equation are defined, the operation on power fuz- zy number and solution of power fuzzy equation based on the structured element method are investigated, and their expressions of membership function are given. At the same time, using the monotonic functions on [-1,1], we replace the quadratic fuzzy equations by a crisp parametric equations and give the necessary and sufficient condition for a fuzzy solution existence, moreover, two numerical examples are designed.Finally, according to the structured element method, solving method of fuzzy linear systems is given, and the necessary and sufficient condition for a fuzzy solution existence is obtained, moreover, a numerical example is given to illustrate our theory. Because the solving method is realized by the Symmetrical monotonic function with y axis on [-1,1], the result show that this solving method is better than the embedding method presented for judging the fuzzy solution existence. At the same time, the paper also investigates a method of solving the fuzzy linear systems which are linear formed by fuzzy structured element, and its advantage that it can be replaced by traditional linear system and it don't need to discuss the parameter linear system.The solving methods of fuzzy equations and fuzzy linear systems put forward in this thesis simplify fuzzy operation and achieve the determination and expression of a fuzzy solution, based on fuzzy structured element theory, which lay the solid foundation for the research of basic fuzzy mathematics theory and the wide application of practical problems.