Optimal Assembly of an s-Stage k-OUT-OF-n System

Consider a system with n components and a reliability function $R(x_1 , \cdots ,x_n )$ where $x_j $ is the reliability of component j. Suppose that each component consists of the same set of parts, say $h_i $ parts of type i for $i = 1, \cdots ,m$, which must all work for the component to work. The problem is to assign $nh_i $ parts of type $i,i = 1, \cdots ,m$, with various part reliabilities to the n components to maximize the system reliability.An assignment is called monotone if all the best parts go to one component, all the next best parts go to the second component, and so on. The optimality of monotone assignments for systems with symmetric and asymmetric reliability functions is studied. In particular, it is proved that monotone assignments are optimal for s-stage k-out-of-n systems.