Many engineering systems can perform their intended tasks with various levels of performance, which are modeled as multi-state systems (MSS) for system availability/reliability assessment problems. Uncertainty is an unavoidable factor in MSS modeling and it must be effectively handled. In this work, we extend the traditional universal generating function (UGF) approach for multi-state system (MSS) availability/reliability assessment to account for both aleatory and epistemic uncertainties. First, a theoretical extension, named hybrid UGF (HUGF), is made to introduce the use of random fuzzy variables (RFVs) in the approach; second, the composition operator of HUGF is defined by considering simultaneously the probabilistic convolution and the fuzzy extension principle; finally, an efficient algorithm is designed to extract probability boxes (p-boxes) from the system HUGF, which allow quantifying different levels of imprecision in system availability/reliability estimation. The HUGF approach is demonstrated on a numerical example and applied to study a distributed generation system, with a comparison to the widely used Monte Carlo simulation method.