From the decision-based design perspective, decision making is the critical element of the design process. All practical decision making occurs under some degree of uncertainty. Subjective expected utility theory is a well-established method for decision making under uncertainty; however, it assumes that the DM can express his or her beliefs as precise probability distributions. For many reasons, both practical and theoretical, it can be beneficial to relax this assumption of precision. One possible means for avoiding this assumption is the use of imprecise probabilities. Imprecise probabilities are more expressive of uncertainty than precise probabilities, but they are also more computationally cumbersome. Probability Bounds Analysis (PBA) is a compromise between the expressivity of imprecise probabilities and the computational ease of modeling beliefs with precise probabilities. In order for PBA to be implemented in engineering design, it is necessary to develop appropriate computational methods for propagating probability boxes (p-boxes) through black box engineering models. This thesis examines the range of applicability of current methods for p-box propagation and proposes three alternative methods. These methods are applied towards the solution of three successively complex numerical examples.