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# Emulation of Utility Functions Over a Set of Permutations: Sequencing Reliability Growth Tasks

Summary: We consider Bayesian design of experiments problems in which we maximize the prior expectation of a utility function over a set of permutations, for example, when sequencing a number of tasks to perform. When the number of tasks is large and the expected utility is expensive to compute, it may be unreasonable or infeasible to evaluate the expected utility of all permutations. We propose an approach to emulate the expected utility using a surrogate function based on a parametric probabilistic model for permutations. The surrogate function is fitted by maximizing the correlation with the expected utility over a set of training points. We propose a suitable transformation of the expected utility to improve the fit. We provide results linking the correlation between the two functions and the number of expected utility evaluations to undertake. The approach is applied to the sequencing of reliability growth tasks in the development of hardware systems, in which there are a large number of potential tasks to perform and engineers are interested in meeting a reliability target subject to minimizing costs and time. An illustrative example shows how the approach can be used and a simulation study demonstrates the performance of the approach more generally. Supplementary materials for this article are available online.

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**Topics:**Statistics**Keywords:**Bayesian design of experiments, Benter model, Design for reliability, Multi-attribute utility**Author:**Wilson, Kevin J.; Henderson, Daniel A.; Quigley, John**Journal:**Technometrics