Optimization Models for Training Belief-Rule-Based Systems

A belief rule-base inference methodology using the evidential reasoning approach (RIMER) has been developed recently, where a new belief rule representation scheme is proposed to extend traditional IF-THEN rules. The belief rule expression matrix in RIMER provides a compact framework for representing expert knowledge. However, it is difficult to accurately determine the parameters of a belief rule base (BRB) entirely subjectively, particularly, for a large-scale BRB with hundreds or even thousands of rules. In addition, a change in rule weight or attribute weight may lead to changes in the performance of a BRB. As such, there is a need to develop a supporting mechanism that can be used to train, in a locally optimal way, a BRB that is initially built using expert knowledge. In this paper, several new optimization models for locally training a BRB are developed. The new models are either single- or multiple-objective nonlinear optimization problems. The main feature of these new models is that only partial input and output information is required, which can be either incomplete or vague, either numerical or judgmental, or mixed. The models can be used to fine tune a BRB whose internal structure is initially decided by experts' domain-specific knowledge or common sense judgments. As such, a wide range of knowledge representation schemes can be handled, thereby facilitating the construction of various types of BRB systems. Conclusions drawn from such a trained BRB with partially built-in expert knowledge can simulate real situations in a meaningful, consistent, and locally optimal way. A numerical study for a hierarchical rule base is examined to demonstrate how the new models can be implemented as well as their potential applications.