Multi-objective evolutionary learning of granularity, membership function parameters and rules of Mamdani fuzzy systems

In this paper, we propose a multi-objective evolutionary algorithm (MOEA) to generate Mamdani fuzzy rule-based systems with different trade-offs between accuracy and complexity by learning concurrently granularities of the input and output partitions, membership function (MF) parameters and rules. To this aim, we introduce the concept of virtual and concrete partitions: the former is defined by uniformly partitioning each linguistic variable with a fixed maximum number of fuzzy sets; the latter takes into account, for each variable, the number of fuzzy sets determined by the evolutionary process. Rule bases and MF parameters are defined on the virtual partitions and, whenever a fitness evaluation is required, mapped to the concrete partitions by employing appropriate mapping strategies. The implementation of the MOEA relies on a chromosome composed of three parts, which codify the partition granularities, the virtual rule base and the membership function parameters, respectively, and on purposely-defined genetic operators. The MOEA has been tested on three real-world regression problems achieving very promising results. In particular, we highlight how starting from randomly generated solutions, the MOEA is able to determine different granularities for different variables achieving good trade-offs between complexity and accuracy.