Nonlinear optimization with fuzzy constraints by multi-objective evolutionary algorithms

Fuzzy constrained optimization problems have been extensively studied since the seventies. In the linear case, the first approaches to solve the so-called fuzzy linear programming problem were made in [12] and [15]. Since then, important contributions solving different linear models have been done and these models have been recipients of a great dealt of work. In the nonlinear case the situation is quite different, as there is a wide variety of specific and both practical and theoretically relevant nonlinear problems, each having a different solution method. In the following we consider a Nonlinear Programming problem with fuzzy constraints. From a mathematical point of view the problem can be addressed as: Min f(x) s.t.:gj (x) bj , j = 1, ¼,m xi Î [li ,ui ], i = 1, ¼,n, li \geqslant 0 \begin{gathered} Min f(x) \hfill \\ s.t.:g_j (x) \lesssim b_j , j = 1, \ldots ,m \hfill \\ x_i \in [l_i ,u_i ], i = 1, \ldots ,n, l_i \geqslant 0 \hfill \\ \end{gathered} (1) where x = (x 1, . . ., x n) ∈ ℜn is a n dimensional real-valued parameter vector, [l i, u i] ⊂ ℜ, b j ∈ ℜ, f (x), g j (x) are arbitrary functions, and the symbol ≲ indicates a fuzzy constraint [15]. Here we will consider the following linear membership function related to each fuzzy constraint: mj (x) = { 0 if gj (x) \geqslant bj + dj h( \tfracbj + dj - gj (x)dj ) if bj \leqslant gj (x) \leqslant bj + dj 1 if gj (x) \leqslant bj \mu _j (x) = \left\{ \begin{gathered} 0 if g_j (x) \geqslant b_j + d_j \hfill \\ h\left( {\tfrac{{b_j + d_j - g_j (x)}} {{d_j }}} \right) if b_j \leqslant g_j (x) \leqslant b_j + d_j \hfill \\ 1 if g_j (x) \leqslant b_j \hfill \\ \end{gathered} \right. (2) which gives the accomplishment degree of g j (x), and consequently of x, with respect to the j-th constraint (the decision maker can tolerate violations of each constraint up to the value b j + d j, j = 1, . . ., m). We assume that the function h is a arbitrary function which allows to represent accurately the accomplishment degree.