Comparison of Probabilistic and Fuzzy Set Methods for Designing under Uncertainty

The objective of this paper is to compare probabilistic models and fuzzy set models for design against uncertainty when there is limited information about the statistics of the uncertainty or modeling error. First, we compare the axioms of probabilistic and fuzzy set methods and the rules governing the arithmetic operations that these methods use. Then, we compare the two methods in designing for maximum safety under a given budget. In general, if there is sufficient information to build accurate probabilistic models of uncertainties, probabilistic methods are better than fuzzy set methods. On the other hand, fuzzy set methods can be better if little information is available. One reason is that it is easier to identify the most conservative fuzzy set model than the most conservative probabilistic model that is consistent with the available information. Introduction Probabilistic models and fuzzy set models primarily describe different aspects of uncertainty. Probabilistic models mainly describe random variability in parameters. In engineering system safety, examples are variability in material properties, geometrical dimensions, or wind loads. In contrast, fuzzy set models of uncertainty mainly describe vagueness, such as vagueness in the definition of safety. When there is only limited information about variability, it is possible to use probabilistic models by making suitable assumptions on the statistics of the variability. However, it has been repeatedly shown that this can entail serious errors (Ben-Haim and Elishakoff, 1990, Neal, et al., 1992). Fuzzy set models, which require little data, appear to be well suited to deal with design under uncertainty when little is known about the uncertainty. Several studies have compared fuzzy set and probabilistic methods in analysis of safety of systems under uncertainty (e.g., Chiang and Dong, 1 Graduate student, 2 Associate Professor, 3 Professor, Fellow AIAA Copyright  1999 by Efstratios Nikolaidis. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission 1987, Hasselman, et al., 1994, Wood, et al., 1990). However, no study has systematically compared the two approaches as a function of the amount of available information. Such comparison, in the context of design against failure, is the objective of the study presented in this paper. For a given system design and a given amount of data about uncertainties, probabilistic analysis yields the probability of failure, a measure that varies between zero and one. Fuzzy set analysis yields a measure called possibility of failure, which also varies between zero and one, with small numbers indicating a higher degree of safety. However, the two measures are not directly comparable. Therefore, to compare the two approaches, we must concentrate on design rather than analysis. That is, with a given amount of resources and a given amount of data about the uncertainty, we use probabilistic methods and fuzzy set methods to obtain two designs. We would like to find which design is more likely to be safer as a function of the amount and accuracy of the available data. The study presented in this paper is limited to problems in which failure is catastrophic. That is, failure is sudden and disastrous. This means the boundary separating failure and success is clear and crisp. Therefore, alternative designs can be ranked by comparing their relative frequencies of failure, which ideally are estimated from experiments. To understand the differences between probabilistic and fuzzy set methods we do the following: a. Compare the axioms of probabilistic and fuzzy set methods. b. Compare fuzzy number calculus with probability calculus. c. Construct and solve simple design problems, in which probabilistic and fuzzy set methods give significantly different results. The following sections describe each task. We use the terms “fuzzy set methods” and “possibility-based methods” interchangeably.