An optimal maintenance model for a combination of secondhand–new or outdated–updated system

In this paper, we study a general maintenance model. Assume that at the beginning, a secondhand (outdated) system is installed, thereafter it will be replaced by a new (updated) system. Two types of compound maintenance policy are used. A policy (t, T) is to replace the secondhand (outdated) system at time t and replace a new (updated) system at time T; whereas a policy (n, N) is to replace the secondhand (outdated) system at the time of nth failure and replace a new (updated) system at the time of Nth failure. We show that an optimal policy (n*, N*) is at least as good as an optimal policy (t*, T*). Furthermore, for a monotone process model, by the semi-Markov decision process approach, an optimal policy (n*, N*) is determined explicitly for maximizing the total expected discounted reward.