No-Free-Lunch Theorems for Reliability Analysis

In most engineering problems, because of a lack of complete information about the structure of the performance function, selection of the optimal approach for efficient reliability analysis is in essence a decision under uncertainty. This issue is investigated in this paper and, by representing reliability methods as search algorithms, no-free-lunch theorems (NFL) of search and optimization are used to propose similar NFL for reliability analysis. Using NFL, this study aims to answer some basic questions about the existence and selection of optimal reliability methods for black- and gray-box problems and proposes a mathematical framework for the application of detection theory in structural reliability. Black- and gray-box problems in this context refer to structural reliability problems with, respectively, no and partial information on the topology of the limit state function. Then, by employing Dempster-Shafer theory of evidence as a generalized Bayesian decision-making theorem, a practical experts-in-the-loop approach for the selection of an optimal reliability method in uncertain conditions is proposed. To meet this aim, providing a step-by-step solution of some selection problem examples, it is shown that knowledge of several experts can be fused into one all-encompassing knowledge representation to reduce the probability of making an error in the selection of an optimal approach for efficient reliability analysis.