Least-cost failure diagnosis in uncertain reliability systems

Abstract

In many textbook solutions, for systems failure diagnosis problems studied using reliability theory and artificial intelligence, the prior probabilities of different failure states can be estimated and used to guide the sequential search for failed components after the whole system fails. In practice, however, both the component failure probabilities and the structure function of the system being examined—i.e., the mapping between the states of its components and the state of the system—may not be known with certainty. At best:, the probabilities of different hypothesized system descriptions, each specifying the component failure probabilities and the system's structure function, may be known to a useful approximation, perhaps based on sample data and previous experience. Cost-effective diagnosis of the system's failure state is then a challenging problem. Although the probabilities of component failures are aleatory, uncertainties about these probabilities and about the system structure function are epistemic. This paper examines how to make best use of both epistemic prior probabilities for system descriptions and the information gleaned from costly inspections of component states after the system fails, to minimize the average cost of identifying the failure state. Two approaches are introduced for systems dominated by aleatory uncertainties, one motivated by information theory and the other based on the idea of trying to prove a hypothesis about the identity of the failure state as efficiently as possible. While the general problem of cost-effective failure diagnosis is computationally intractable (NP-hard), both heuristics provide useful approximations on small to moderate sized problems and optimal results for certain common types of reliability systems, including series, parallel, parallel-series, and k-out-of- n systems. A hybrid heuristic that adaptively chooses which heuristic to apply next after any sequence of observations (component test results) appears to give excellent results. Several computational experiments are summarized in support of these conclusions, and extensions to reliability systems with repair are briefly considered. Next, it is shown that diagnosis can proceed when aleatory and epistemic uncertainties are both present using the same techniques developed for aleatory probabilities alone. If only the epistemic probability distribution of system descriptions is known, then the same heuristics that are used to diagnose a system's failure state for systems with known descriptions can also be used to identify the system and diagnose its failure state when there is epistemic uncertainty about the identity of the system. This result suggests a unified approach to least-cost failure diagnosis in reliability systems with both aleatory probabilities of component failures and epistemic probabilities for system descriptions.