Numerical Solution of a System of Polynomial Parametric form Fuzzy Linear Equations

Since fuzzy logic was introduced by Lotfi Zadeh in 1965 (41), it has had many successful applications in all fields that one can imagine. The reason is that many real-world applications problems are involved the systems in which at least some parameters are represented by fuzzy numbers rather than crisp numbers and linguistic labels such as small and large are also associated with the fuzzy sets. On the other hand a system of fuzzy linear equations may appear in a wide variety of problems in various areas such as mathematics, statistics, physics, engineering and social sciences. The objective of this chapter is to introduce a method to find a good approximate solution to a system of fuzzy linear equations, and first we need to be familiar with some notations on fuzzy numbers in this chapter, however it is assumed that the reader is relatively familiar with the elementary fuzzy logic concepts. In (14), Chong-Xin and Ming represented a fuzzy number ũ by an ordered pair of functions (u(r),u(r)), 0≤ r ≤ 1, which satisfies some requirements. In papers and books the authors mainly have used linear membership functions as spreads, because they are conceptually the simplest, have a clear interpretation and play a crucial role in many areas of fuzzy applications, and almost every works on this field of study have been done on triangular or trapezoidal fuzzy numbers, but polynomial form fuzzy numbers are simple and have a clear interpretation too, and in order to obtain a richer class of fuzzy numbers we use polynomials of the degree higher than one, as the spreads of membership functions of fuzzy numbers. Thus in (2) we introduced a type of fuzzy numbers in which both left spread function u(r) and right spread function u(r) are polynomials of degree at most m. We named this type of fuzzy numbers, m-degree polynomial-form fuzzy numbers. The main aim of introducing this type of fuzzy numbers is that in many applications of fuzzy logic and fuzzy mathematics we need (or it is better) to work with the same fuzzy numbers. It has been shown in (2) that a fuzzy number ũwith continuous left and right spread functions can be approximated by a fuzzy number with m-degree polynomial-form, where choosing m depends on the shape of left and right spread functions L and R, and the derivation order of them. Some applications of this approximation in the case m = 1 (trapezoidal fuzzy numbers) are given in (3), and some properties of this approximation operator are recently given in (10). There are many other literatures which authors tried to approximate a fuzzy number by a simpler one (2; 3; 4; 18; 25; 27; 28; 29; 30; 31; 32; 42). Also there are some distances defined by authors to compare fuzzy numbers (35; 37). 23 4