The primary object of this research was to develop a two-phase strategy for preventive maintenance and the critical number of units. Maintenance is done if the number of failures is equal to or greater than the critical number of units. The system under consideration had q components. The system failed when k (k < q) components or more failed. This system when subjected to preventive maintenance can be described by a Markov Process. The transition probabilities of the Markov Process were obtained from the distributions qf the strength and stress of the components. The underlying distributions were assumed to be double exponential. Various combinations of the inspection period and the critical number of units were used to obtain the global minimum. The criteria were that the system should not fail and minimum cost. / Master of Science