We consider the problem of determining the optimal sequence of tests for the discovery of a faulty component, where there is a random cost associated with testing a component. Our work is motivated by applications in telecommunications networks, e,g,, location and isolation of faults (or intruders) in IP networks. A novel feature in our approach is that a risk-sensitive performance criterion is used in order to rank different competing schedules. Risk-sensitivity is incorporated through the use of an exponential utility function, and hence optimal schedules attain a trade-off between minimal expected costs and, e,g., a low variance about the achievable expected costs. We characterize optimal schedules both when the testing sequence is not subject to precedence constraints, and when it is subject to such constraints, given by an arbitrary partial order. For the case with precedence constraints, we show that our models can be analyzed via modular decompositions, as studied by Monma and Sidney.