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This paper analyzes the importance of components within a multiple-state uncertain random k -out-of- n system, which is called ur-m- k -out-of- n system for short. This system simultaneously contains randomness and uncertainty. The states of components exhibit polymorphism and are defined as uncertain random variables. The component importance analysis is both a prerequisite and a foundation for system reliability optimization. For this reason, the marginal reliability importance (MRI) and joint reliability importance (JRI) are proposed to examine the importance of an individual component or couple components of ur-m- k -out-of- n systems using chance theory. Some theorems are given to calculate MRI and JRI. Moreover, this paper gives several numerical examples to illustrate how to calculate MRI and JRI. • Propose component importance for a multi-state uncertain random k -out-of- n system. • Propose two-component joint importance for the above system. • Provide some formulas to calculate the importance and joint importance. [ABSTRACT FROM AUTHOR]

• Information entropy is first used for Kriging-based system reliability assessment. • The prediction uncertainty of the system state is quantified by information entropy. • A learning function is proposed using the prediction uncertainty of the system state. • Sample with the largest uncertainty get more attention for model reconstruction. Structural reliability assessment is a popular topic in engineering problems, particularly for the larger and more complex systems with implicit performance functions, also called black-box problems. The reliability assessment for a black-box problem must continuously be computed using simulation models, such as the finite element model, a highly time-consuming process with a high computational cost. The adaptive Kriging has gained considerable attention over the past decade. The Kriging-based reliability assessment method reduces the computational cost to a great extent, on the premise of ensuring the accuracy of reliability assessment. However, many of the currently published system reliability assessment methods, construct adaptive Kriging models by reducing the probability of incorrect prediction of the system state, and do not make full use of the uncertainty of the system state prediction information. To this end, a new Kriging-based method for structural system reliability assessment is proposed in this study. First, the probabilities of incorrect or correct system state predictions were understood from the perspective of information entropy. Second, an active learning strategy is proposed based on information entropy theory. Finally, the advantages of the proposed method are demonstrated and highlighted through several numerical examples. The results show that the proposed method achieves a good balance between the accuracy and computational cost, and the numerical magnitude effect does not affect the computational cost. Moreover, this is an effective method for assessing the reliability of complex systems. [ABSTRACT FROM AUTHOR]