A Genetic Algorithm for Linear Programming With Fuzzy Constraints

We examine a linear programming problem formulation in which the constraint coefficients are not precisely given in the work. We investigate the possibility of applying GAs to solve this kind of fuzzy linear programming problem without defining membership functions for fuzzy numbers, using the extension principle, interval arithmetic, and α-cut operations for fuzzy computations, and using a penalty method for constraint violations. The proposed approach simulates every fuzzy number by distributing it into certain partition points. GAs are then used to evolve the values in each partition point. As a result, the final values represent the membership grade of that fuzzy number. After calculating the estimated values of each uncertain coefficient, we obtain a defuzzified linear programming problem. The crisp problem can then be solved using the following GA stage. The empirical results show that the proposed approach can obtain very good solutions within the given bounds for each fuzzy coefficient, thereby accomplishing flexible linear programming.