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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]

This paper presents an uncertain random multi-state k -out-of- n system (urm- k -out-of- n system, for short), where both random components and uncertain components coexist. States of the system and components display multiplicity and are defined as uncertain random variables. Since reliability is a crucial index to reflect system performance, this paper proves some formulas to calculate the reliability of urm- k -out-of- n systems in accordance with chance theory, which have two special cases, random systems and uncertain systems. Later, a mixed algorithm for multi-valued decision diagram (MDD) and binary search is designed to assess the reliability index. Compared to the general algorithm, it operates in significantly less time when the system becomes complex. [ABSTRACT FROM AUTHOR]