Finding solutions of fuzzy polynomial equations systems by an Algebraic method

Computing real solutions of fuzzy polynomial systems

“…An easy computation shows the equivalence of the systems T 1 and T (F 1 ), whose set of solutions is {(x, y) ∈ R | x y = 1}. This phenomenon of equivalence between both approaches may be explained in a very general way as we show below by considering the classical computation of the collected crisp form obtained by an application of the algorithm of [20] on the generic equation ( 5) of (E).…”

“…Our approach allows at the same time to improve and to generalize the methods known so far. For instance, results in [18] and [20] are restricted to triangular fuzzy numbers. Indeed, the crisp form of (E) with two parameters u = L −1 (r ) and v = R −1 (r ) given in Identity ( 9) is a generalization of the crisp form known in triangular case with only one parameter r where r ∈ [0, 1].…”

“…In the specific case of triangular fuzzy numbers as coefficients, the authors of [18] and [20] compute a collected crisp form of (S) formed by 4 s real algebraic equations. In this part we are interested in the relationship between their collected crisp form with 4 s equations and our collected crisp system with 3 s equations, that is the real transform of (S).…”

“…Applied to first equation, the algorithm proposed in [20] produces the following collected crisp form:…”

“…Recently, a global approach using classical algebraic techniques has been developed [18,19,20,21]. Indeed, despite a name that may be confusing, fuzzy numbers benefit from a perfectly formal definition.…”