Opportunistic Replacement of a Single Part in the Presence of Several Monitored Parts

In a system with several stochastically failing parts and economies of scale in their maintenance, it may be advantageous to follow an "opportunistic" policy for maintenance. In opportunistic policies the action to be taken on a given part at a given time depends on the state of the other parts of the system. For the model considered in this paper, it is assumed that all parts of the system but one are inspected continuously (monitored) and that the remaining part cannot be inspected except when it is replaced. If the monitored parts have exponential distributions of time to failure, the optimal replacement policy for the remaining part has the following form: Let the non-monitored part be labeled 0 and let there be M monitored parts, labeled 1, ..., M; then there are M + 1 numbers n 1 , ..., n M , N, with 0 \leqq n i \leqq N \leqq \infty such that: (a) if Part i fails at a time when the age of Part 0 is between 0 and n i , replace Part i alone (i - 1, ..., M); (b) if Part i fails at a time when the age of Part 0 is between n i and N, then replace Parts 0 and i together; (c) if Part 0 reaches age N at a time when all monitored parts are good, replace Part 0 alone. It is demonstrated that the parameter N is finite provided that the reliability of the non-monitored part approaches zero as the age of this part goes to infinity. These results are proved for several alternative criteria for evaluating replacement policies. Computation of the parameters of the optimal policy is also discussed. Applications of this policy are discussed in [Jorgenson, D., J. J. McCall. 1962. Optimal Scheduling of Replacement and Inspection. The RAND Corporation, Memorandum RM-3098-PR, May.].