As a rule-based expert system, the belief rule base (BRB) exhibits tremendous advantages in modeling nonlinearity for complex systems. Present BRB learning studies can be classified into three categories: BRB structure learning, BRB parameter learning, and BRB joint optimization but only in an iterative and separate fashion. In this study, a novel Parallel Multipopulation optimization approach for BRB, i.e., PMP-BRB, is proposed that simultaneously optimizes the structure and parameters of a BRB. In the optimization model of PMP-BRB, the structure of BRB, i.e., the number of rules, is used as another decisive variable. In the optimization algorithm of PMP-BRB, multiple populations are initialized and subsequently optimized, with different populations representing BRBs of varied sizes. Furthermore, a “completion and deletion” strategy is proposed, wherein individual BRBs in different populations are completed with additional genes that only engage in the optimization operations but not in fitness calculations. Moreover, a trade-off analysis is conducted for decision-makers to identify the final optimal configuration of a BRB based on their preference. The proposed PMP-BRB approach is validated by four cases, namely a numerical case, two practical cases, and a case study with five classification benchmarks. Four evolutionary algorithms are tested and compared. With the structure and parameters optimized simultaneously, all the four cases yield competitive results in comparison with those of previous studies.

作为基于规则的专家系统，信念规则库（BRB）在建模复杂系统的非线性方面显示出巨大的优势。当前的BRB学习研究可以分为三类：BRB结构学习，BRB参数学习和BRB联合优化，但只能以迭代和单独的方式进行。在这项研究中，提出了一种新颖的针对BRB的并行多种群优化方法，即PMP-BRB，它同时优化了BRB的结构和参数。在PMP-BRB的优化模型中，BRB的结构（即规则数量）被用作另一个决定性变量。在PMP-BRB的优化算法中，多个种群被初始化并随后被优化，其中不同的种群代表大小不同的BRB。此外，提出了“完成和删除”策略，其中，不同种群中的个体BRB用仅参与优化操作而不参与适应度计算的其他基因完成。此外，还为决策者进行了权衡分析，以根据他们的偏好确定BRB的最终最佳配置。所提出的PMP-BRB方法通过四个案例进行了验证，即一个数值案例，两个实际案例以及一个具有五个分类基准的案例研究。测试并比较了四种进化算法。在同时优化结构和参数的情况下，与以前的研究相比，这四个案例均具有竞争优势。进行权衡分析，以便决策者根据他们的偏好确定BRB的最终最佳配置。所提出的PMP-BRB方法通过四个案例进行了验证，即一个数值案例，两个实际案例以及一个具有五个分类基准的案例研究。测试并比较了四种进化算法。在同时优化结构和参数的情况下，与以前的研究相比，这四个案例均具有竞争优势。进行权衡分析，以便决策者根据他们的偏好确定BRB的最终最佳配置。所提出的PMP-BRB方法通过四个案例进行了验证，即一个数值案例，两个实际案例以及一个具有五个分类基准的案例研究。测试并比较了四种进化算法。在同时优化结构和参数的情况下，与以前的研究相比，这四个案例均具有竞争优势。