Expected cost comparisons for inspection and repair policies

An (n − r+ 1) out of n system is down (at time t0, say) due to the failure of at least r of its components. Each component is made up of t equally reliable parts assembled in parallel. All parts are assumed to function independently of one another. Three different policies are described for inspecting and repairing the parts of one component after another until all (or some) of the failed components are identified (and fixed). We assume that a less reliable part is cheaper to repair than a more reliable one. We show that in order to minimize the expected total cost of necessary repairs, for each of the three policies, we should inspect the weakest component first, then the second weakest and so on. We also show that the closer we follow the optimal order (in a specific sense described by a partial ordering b2 on the set of permutations of {1,…, n}) the less is the expected cost that we incur.