  We consider Bayesian design of experiments problems in which we maximise 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 maximising 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 is a large number of potential
tasks to perform and engineers are interested in meeting a reliability target
subject to minimising 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.
