Weighted Constraints in Fuzzy Optimization

Many practical optimization problems are characterized by some flexibility in the problem constraints, where this flexibility can be exploited for additional trade-off between improving the objective function and satisfying the constraints. Especially in decision making, this type of flexibility could lead to workable solutions, where the goals and the constraints specified by different parties involved in the decision making are traded off against one another and satisfied to various degrees. Fuzzy sets have proven to be a suitable representation for modeling this type of soft constraints. Conventionally, the fuzzy optimization problem in such a setting is defined as the simultaneous satisfaction of the constraints and the goals. No additional distinction is assumed to exist amongst the constraints and the goals. This report proposes an extension of this model for satisfying the problem constraints and the goals, where preference for different constraints and goals can be specified by the decision-maker. The difference in the preference for the constraints is represented by a set of associated weight factors, which influence the nature of trade-off between improving the optimization objectives and satisfying various constraints. Simultaneous weighted satisfaction of various criteria is modeled by using the recently proposed weighted extensions of (Archimedean) fuzzy t-norms. The weighted satisfaction of the problem constraints and goals are demonstrated by using a simple fuzzy linear programming problem. The framework, however, is more general, and it can also be applied to fuzzy mathematical programming problems and multi-objective fuzzy optimization.