Multivariate stochastic comparisons of inspection and repair policies

For a system consisting of n independent components, inspection and repair policies are compared in the sense of the usual multivariate stochastic order with respect to the partial orderings [greater-or-equal, slanted]b1 and [greater-or-equal, slanted]b2 on the set of permutations of {1,...,n}. A given permutation [pi] of {1,...,n} determines the order in which components are visited and inspected. We assume that the reliability of the ith component is given by Pi, where P1[less-than-or-equals, slant]P2[less-than-or-equals, slant]...[less-than-or-equals, slant]Pn. If [pi] and [pi]' are two permutations such that [pi][greater-or-equal, slanted]b1[pi]', we show that inspecting the system with [pi] is better than with [pi]' in the sense that more failed components are encountered or less time is used in completing an inspection. If each component is made up of t parts assembled in parallel, it is shown that, for three types of repair policies, the rank of preference of inspection procedures is consistent with the partial ordering [greater-or-equal, slanted]b2 on the permutations. Our main results strengthen those in Boland, El-Neweihi and Proschan (Ann. Appl. Probab. 1 (1991) 207).