Periodic Inspection Optimization of a k-Out-of-n Load-Sharing System

In this paper, we consider a k-out-of-n load-sharing system with n identical components sharing a certain amount of load. Each time a component fails, its load is distributed to the remaining components; we assume an increase in load increases the hazard rates of the remaining components. The system is periodically inspected to detect failed components. Two cases may occur in an inspection interval: if the number of failed components is less than n-k+1, then the failed components are only rectified at periodic inspections; if the number of failures reaches n-k+1, then the system fails, and at this time, all the failed components are inspected and rectified. A failed component is replaced or minimally repaired according to a probability which depends on its age at the failure time. The components' failures follow a Non-Homogenous Poisson Process (NHPP), and their intensity functions depend on their ages and the loads to which they are exposed at any moment. In this paper, we develop a model to find the optimal inspection interval for such a system, which minimizes the total expected cost incurred over the system lifecycle. We derive the analytical solution for the special case of a 1-out-of-2 system, and discuss its computational difficulties. We then present a simulation algorithm to find the required expected values in the objective function. Several numerical examples are presented to illustrate the proposed model.