Optimal Assignment of Components to a Two-Stage k-Out-of-n System

A two-stage k-out-of-n system is an l-out-of-m system except that each of the m parts is itself a k_i-out-of-n_i system. The problem is to assign a set of ∑_i=1^mn_i probabilities to the ∑_i=1^mn_i components in the system to maximize its reliability. In this paper we consider the case k_i = k for all i. We give optimal assignments when the first stage is a parallel system or the second stage is a series system. We conjecture that the optimal assignment problems for all other two-stage k-out-of-n systems are NP-complete, but we are able to prove this only when the second stage is a parallel system. We also propose a heuristic assignment algorithm for the general case.