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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]