A RULE-BASED METHOD TO CALCULATE THE WIDEST SOLUTION SETS OF A MAX-MIN FUZZY RELATIONAL EQUATION

In this paper, we propose a new rule-based method to determine exactly all the widest solution sets of a max-min fuzzy relational equation. As they can be applied to many fields of fuzzy logic, fuzzy relational equations are of great importance; therefore, many researches have been carried out on the subject and many methods have already been proposed to solve the various kinds of fuzzy equations that can be defined, especially the so-called max-min equations. One of the main consequences of the use of the max and min operators is the non uniqueness of the solution. Indeed, when a max-min fuzzy equation has solutions, there are several solutions which can be grouped in intervals. However, most of the existing methods do not enable to calculate the widest solution sets only, often leading to calculate non-maximal solution sets, that is, solution sets included in wider sets. This often results in unnecessary calculations, requiring huge amounts of memory and time. The method we propose here enables to determine rapidly all the widest solution sets of a max-min equation, and those sets only. The use of symbol-valued matrices enables to define simple rules and makes the method very simple to apply.