Optimal replacement policy for a two-unit system subject to shocks and cumulative damage

Abstract

This paper investigates an extended replacement policy for a two-unit system subject to shocks and cumulative damage. As a shock occurs, the system suffers two types of effects: type I shock-effect is rectified by a minimal repair and type II shock-effect is removed by a corrective replacement. The probability of type II shock depends on the number of shocks from the last replacement. Each type I shock causes a minor failure of unit 1 and some damage to unit 2. These damages to unit 2 are additive and the system fails when the total damage exceeds a critical level L. A replacement policy (T, m, l) is considered in which the system is preventively replaced at age T or at the mth type I shock or at the time which the total damage to unit 2 exceeds a pre-specified level l; and is correctively replaced either at the first type II shock or when the total damage to unit 2 exceeds a failure level L, whichever occurs first. The optimal policy that minimizes the expected cost rate is determined analytically. The proposed model extends several existing results. Our replacement policy is more general, flexible, and applicable in the real situations.