Developing and optimizing fuzzy relation equations are of great relevance in system modeling, which involves analysis of numerous fuzzy rules. As each rule varies with respect to its level of influence, it is advocated that the performance of a fuzzy relation equation is strongly related to a subset of fuzzy rules obtained by removing those without significant relevance. In this study, we establish a novel framework of developing granular fuzzy relation equations that concerns the determination of an optimal subset of fuzzy rules. The subset of rules is selected by maximizing their performance of the obtained solutions. The originality of this study is conducted in the following ways. Starting with developing granular fuzzy relation equations, an interval-valued fuzzy relation is determined based on the selected subset of fuzzy rules (the subset of rules is transformed to interval-valued fuzzy sets and subsequently the interval-valued fuzzy sets are utilized to form interval-valued fuzzy relations), which can be used to represent the fuzzy relation of the entire rule base with high performance and efficiency. Then, the particle swarm optimization (PSO) is implemented to solve a multi-objective optimization problem, in which not only an optimal subset of rules is selected but also a parameter ε for specifying a level of information granularity is determined. A series of experimental studies are performed to verify the feasibility of this framework and quantify its performance. A visible improvement of particle swarm optimization (about 78.56% of the encoding mechanism of particle swarm optimization, or 90.42% of particle swarm optimization with an exploration operator) is gained over the method conducted without using the particle swarm optimization algorithm.

中文翻译：

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基于数据子集的粒状模糊关系方程的发展


开发和优化模糊关系方程在系统建模中具有重要意义，系统建模涉及大量模糊规则的分析。由于每个规则的影响程度各不相同，因此建议模糊关系方程的性能与通过删除那些不具有显着相关性的规则而获得的模糊规则子集密切相关。在这项研究中，我们建立了一个开发粒状模糊关系方程的新颖框架，该框架涉及模糊规则的最佳子集的确定。通过最大化所获得的解决方案的性能来选择规则子集。本研究的独创性是通过以下方式进行的。从开发粒状模糊关系方程开始，基于选定的模糊规则子集确定区间值模糊关系（将规则子集转换为区间值模糊集，然后利用区间值模糊集形成区间值） -值模糊关系），可以高性能、高效地表示整个规则库的模糊关系。然后，实现粒子群优化（PSO）来解决多目标优化问题，其中不仅选择规则的最优子集，而且确定用于指定信息粒度级别的参数ε。进行了一系列实验研究来验证该框架的可行性并量化其性能。与不使用粒子群优化算法的方法相比，粒子群优化的方法得到了明显的改进（粒子群优化的编码机制大约提高了 78.56%，或者带有探索算子的粒子群优化的 90.42%）。