An optimal inspection policy for a standby system under limited inspection opportunities

Inspection due to its relatively simple implementation is increasingly used in industry to identify the system state, to reveal system fault, and to make maintenance decisions. Researchers and engineers have almost always investigated optimal inspection policies for standby systems under an assumption that the servicing systems fail following exponential distributions. In many cases, however, the exponential distribution assumption can restrict the applicability of inspection policies due to its memory-less nature. In this paper, we consider the optimal inspection problem of a stochastically failing warm-standby system that will be needed at a random time when a failure of the main servicing system occurs. If the warm-standby system is not operational at that time, the system incurs a large penalty, which we want to avoid through inspections. To address this issue, we investigate an optimal inspection policy where the failure of the servicing system follows Weibull distribution. A model to schedule limited optimal inspection times is presented so that the total cost consisting of the expected penalty and inspection costs is minimized. The structural properties of the optimal policy are discussed in detail, and a solution algorithm is presented to find the optimal inspection times, which can guarantee the existence and uniqueness of the optimal policy. Thus, the commonly used inspection policies limited to the exponential cases turn out to be our special cases. Numerical simulations are provided to demonstrate the proposed approach.