Appropriate uncertainty models are required for realistic representations of quantities in real world engineering tasks. Uncertainty quantification is applied to estimate the uncertainty of system responses, with respect to uncertain input quantities. In contrast to aleatoric uncertainty, which is based on natural variability, epistemic uncertainty is caused by lack of knowledge, incertitudes or inaccuracy. In this contribution, epistemic uncertainties are modeled by fuzzy quantities and corresponding uncertainty quantification approaches are investigated. The propagation of fuzzy quantities is based on the extension principle. For numerical analyses, a discretization of the extension principle is required, which can be reformulated as an optimization problem. Two different approaches are state-of-the-art for formulating the optimization problem of the extension principle, which are referred to as α-level optimization and sampling-based approach (SBA). A comparison of these two approaches is presented, highlighting their advantages and deficits with respect to efficiency and accuracy of the fuzzy analyses. Based on the advantages of both α-level optimization and SBA, a novel approach, the intermediately discretized extended α-level optimization (IDEALO), is developed. In IDEALO, advantages of α-level optimization and SBA are combined to a hybrid approach. The superiority of IDEALO over the other two approaches is demonstrated in benchmark examples.