Reliability estimation and design with insufficient data based on possibility theory

Early in the engineering design cycle, it is difficult to quantify product reliability or compliance to performance targets because of insufficient data or information for modeling the uncertainties. Design decisions are, therefore, based on fuzzy information that is vague, imprecise, qualitative, linguistic, or incomplete. The uncertain information is usually available as intervals with lower and upper limits. In this work, the possibility theory is used to assess design reliability with incomplete information. The possibility theory can be, viewed as a variant of fuzzy set theory. The formal theories to handle uncertainty are first introduced using the theoretical fundamentals of fuzzy measures. A computationally efficient and accurate hybrid (global-local) optimization approach is subsequently described for calculating the confidence level of fuzzy response. The method combines the advantages of the commonly used vertex and discretization methods. Subsequently, the possibility theory is used in design. A possibility-based design optimization method is proposed where all design constraints are expressed in a possibilistic way. It is shown that the method gives a conservative solution compared with all conventional reliability-based designs obtained with different probability distributions. Finally, a general possibility-based design optimization method, which handles a combination of random and possibilistic design variables, is presented. Several numerical examples demonstrate the application of possibility theory in design.