A trivariate optimal replacement policy for a deteriorating system based on cumulative damage and inspections

In this article, we study a trivariate replacement model for a deteriorating system consisting of two units. Failures of unit 1 can be classified into two types. Type I failure (minor failure) is fixed by a minimal repair and type II failure (catastrophic failure) is removed by a replacement. Both types of failures can only be detected through inspection. Each type I failure of unit 1 will result in a random amount of damage to unit 2 and the damages are cumulative. The probability of type I failure or type II failure is assumed to depend on the number of failures since the last replacement. We formulate a replacement policy based on the number of type I failure, the occurrence of the first type II failure, and the amount of accumulative damages. Hence the system is replaced either preventively or correctively at any of the following four conditions depend on whichever occurs first; preventively (a) at the Nth type I failure; or (b) when the total damage of unit 2 exceeds a pre-specified level Z (but less than the failure level l); and, correctively (c) at the first type II failure; or (d) when the total damage of unit 2 exceeds a failure level l, where Z and l represent the thresholds of total damage level for unit 2 to preventive and corrective replacements, respectively. Although a type I failure can be fixed by a minimal repair, but the operating period is stochastically decreasing and repair time is stochastically increasing as time goes on. The minimal total expected long-run net cost per unit time of the system is derived and a computational algorithm for determining the optimal policy is developed. A real-world application from electric power industry is provided. Several past studied are shown to be special cases of our model. Finally, a numerical example is presented.