A bivariate optimal replacement policy with cumulative repair cost limit for a two-unit system under shock damage interaction

Abstract

In this paper, a bivariate $(n,k)$ replacement policy with cumulative repair cost limit for a two-unit system is studied, in which the system is subjected to shock damage interaction between units. Each unit 1 failure causes random damage to unit 2 and these damages are additive. Unit 2 will fail when the total damage of unit 2 exceed a failure level $K$, and such a failure makes unit 1 fail simultaneously, resulting in a total failure. When unit 1 failure occurs, if the cumulative repair cost till to this failure is less than a predetermined limit $L$, then unit 1 is corrected by minimal repair, otherwise, the system is preventively replaced. The system is also replaced at the $n$th unit 1 failure, or at damage level $k$ (${<}K$) of unit 2, or at total failure. The explicit expression of the long-term expected cost per unit time is derived and the corresponding optimal bivariate replacement policy can be determined analytically or numerically. Finally, a numerical example is given to illustrate the theoretical results for the proposed model.