Solving Linear Optimization Problem with Max-Archimedean Interval-Valued Fuzzy Relation Equations as Constraints

This paper introduces a linear optimization problem subject to max-Archimedean interval-valued fuzzy relation equations. According to the literature, three types of solution sets, namely; tolerable solution set, united solution set and controllable solution set can be identified with interval-valued fuzzy relation equations. Since the tolerable solutions are very useful in fuzzy control problems, thus optimization with such type of fuzzy relation equations is an important topic of research. The structure and the properties of the tolerable solution set are studied. The tolerable solution set can be characterized by one maximum solution and finitely many minimal solutions. Generally, determining all minimal solutions is a computationally difficult task, thus an efficient algorithm based on the rules of reduction is proposed which directly computes the tolerable optimal solution of the problem without finding the set of all minimal solutions. The concept of reduction is efficient for large size problems in terms of computation. The proposed method is illustrated with some examples.