An (n - r + 1) out of n system is down (at time t(o), 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 aft
er 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 expec
ted total cost of necessary repairs, for each of the three policies, w
e should inspect the weakest component first, then the second weakest
and so on. We also show that the closer we follow the optimal order (i
n a specific sense described by a partial ordering greater than or equ
al to(b2) on the set of permutations of {1,..., n}) the less is the ex
pected cost that we incur.