Comparison of Probability and Possibility for Design Against Catastrophic Failure Under Uncertainty

Abstract

This paper compares probabilistic and possibility-based methods for design under uncertainty. It studies the effect of the amount of data about uncertainty on the effectiveness of each method. Only systems whose failure is catastrophic are considered, where catastrophic means that the boundary between success and failure is sharp. First, the paper examines the theoretical foundations of probability and possibility, focusing on the impact of the differences between the two theories on design. Then the paper compares the two theories on design problems. A major difference between probability and possibility is in the axioms about the union of events. Because of this difference, probability and possibility calculi are fundamentally different and one cannot simulate possibility calculus using probabilistic models. Possibility-based methods tend to underestimate the risk of failure of systems with many failure modes. For example, the possibility of failure of a series system of nominally identical components is equal to the possibility of failure of a single component. When designing for safety, the two methods try to maximize safety in radically different ways and consequently may produce significantly different designs. Probability minimizes the system failure probability whereas possibility maximizes the normalized deviation of the uncertain variables from their nominal values that the system can tolerate without failure. In contrast to probabilistic design, which accounts for the cost of reducing the probability of each failure mode in design, possibility tries to equalize the possibilities of failure of the failure modes, regardless of the attendant cost. In many safety assessment problems, one can easily determine the most conservative possibilistic model that is consistent with the available information, whereas this is not the case with probabilistic models. When we have sufficient data to build accurate probabilistic models of the uncertain variables, probabilistic design is better than possibility-based design. However, when designers need to make subjective decisions, both probabilistic and possibility-based designs can be useful. The reason is that large differences in these designs can alert designers to problems with the probabilistic design associated with insufficient data and tell them that they have more flexibility in the design than they may have known.