Optimal Ordering and Replacement Scheduling for a Deteriorating System Subject to Shocks

This paper investigates a spare unit ordering and replacement policy for a deteriorating system with shocks in seeking for the optimal number of minimal repairs. The original system suffers from both deterioration and external shocks, in which the shocks are divided into two distinct categories with different time-dependent probabilities, including a non-fatal shock whose damage can be completely removed by a minimal repair and a fatal shock which can also break down the system. To be specific, the original system failure is subject to a competing failure process, where it occurs at the deteriorating failure time or at the first appearance epoch of a fatal shock. A spare unit with a constant delivery time is ordered emergently at system failure or preventively when the number of minimal repairs reaches a value, whichever occurs first, for the sake of minimizing the long run average cost rate in one renewal cycle. The optimal number of minimal repairs is theoretically demonstrated and an illustrative example is designed as a validation of the theoretical results, as well as sensitivity analyses of some key parameters.