Hybrid uncertainty model for multi-state systems and linear programming-based approximations for reliability assessment

This article studies reliability assessment for Multi-State Systems (MSSs) with components states that are uncertain in both probability and performance realizations. First, we propose a model of (discrete) Hybrid Uncertainty Variable (HUV) for modeling the hybrid uncertainty of the MSS, in which both state performance levels and associated probability level are described by uncertain values. The HUV can be regarded as a generalization of random variable whose realizations and corresponding probabilities are both uncertain values. Especially the uncertain probabilities are controlled by the probability law. Leveraging the HUV-based hybrid uncertainty model, the primitive probability law is considered throughout the whole process from modeling component state probabilities, through the resulting system state probabilities, to the final reliability computations. Therefore, the information loss is reduced to a minimum. Furthermore, we develop a framework for assessing the reliability of the MSS with hybrid uncertainty. In particular, due to hybrid uncertainty considered together with the primitive probability law constraints, the reliability bound computations essentially require solving a pair of multi-linear optimization problems, which in general are non-convex and non-concave and therefore belong to a class of difficult optimization problems. Therefore, we develop a linear programming‐based cut-generation approach for solving the reliability bound assessment problem which achieves a computationally attractive approximation. Finally, the effectiveness of our approaches is validated in the case study with the comparisons to the published results.