Fuzzy reasoning for solving fuzzy mathematical programming problems

We interpret fuzzy linear programming (FLP) problems with fuzzy coefficients and fuzzy inequality relations as multiple fuzzy reasoning schemes (MFR), where the antecedents of the scheme correspond to the constraints of the FLP problem and the fact of the scheme is the objective of the FLP problem. Then the solution process consists of two steps: first, for every decision variable x ∈ IRn, we compute the (fuzzy) value of the objective function, MAX(x), via sup-min convolution of the antecedents/constraints and the fact/objective, then an (optimal) solution to FLP problem is any point which produces a maximal element of the set {MAX(x) | x ∈ IRn} (in the sense of the given inequality relation). We show that our solution process for a classical (crisp) LP problem results in a solution in the classical sense, and (under well-chosen inequality relations and objective function) coincides with those suggested by Buckley [2], Delgado et al. [5, 6], Negoita [9], Ramik and Rimanek [12], Verdegay [16, 17] and Zimmermann [24]. Furthermore, we show how to extend the proposed solution principle to non-linear programming problems with fuzzy coefficients. We illustrate our approach by some simple examples.