Optimal allocation of s-identical, multi-functional spares in a series system

This paper considers the problem of allocating statistically-identical, multi-functional spares to subsystems of a series system. The objective is to maximize the system reliability for mission time T which can be deterministic or stochastic. Several problems which are conceptually similar to this one have been discussed in the literature in different contexts. An algorithm is provided for obtaining standby redundancy allocation, and sufficient conditions are derived for optimality of the resulting allocation for general T. The algorithm is equivalent to a simple allocation rule under the sufficient conditions. The allocation rule gives an optimal allocation for the special cases: the PDF's of component lifetimes are log concave (which implies increasing failure rate), and T is deterministic; the components have exponential failure times, and T follows a gamma distribution; and component lifetime distributions are general, and T follows an exponential or a mixture of exponential distributions. No simpler method is available for latter two cases.