Qualitative properties of profit-making k-out-of-n systems subject to two kinds of failures

Several properties are derived for optimal k-out-of-n systems in which: (1) the i.i.d components can be with a prespecified frequency, in one of two possible modes; (2) components are subject to failures in each of the two modes; and (3) the costs of two kinds of system failures are not necessarily the same. A characterization of the optimal k that maximizes the mean system profit is obtained. It is shown how one can predict directly from the parameters of the system whether the optimal k is smaller or larger than n/2. The directions of change in the optimal k resulting from changes in system parameters are ascertained. A subclass of the formulation and results corresponds to the case examined in the literature in which the optimal k is chosen to maximize the system´s reliability