Joint optimal maintenance and inspection for a k-out-of-n system

In the present paper, a k-out-of-n system with hidden failures is considered. The system is inspected periodically over a finite lifecycle. Hidden component failures accumulate and cause system failure when their cumulative number reaches n - k + 1. Every system failure presents an additional opportunity for inspection and, therefore, is called "opportunistic". The objective is to find the optimal periodic inspection policy and the optimal maintenance action at each inspection for the entire system. Three types of maintenance are considered: minimal repair, preventive replacement and corrective replacement. In view of the failures being hidden, the maintenance decision is based on the optimal number of minimal repairs before replacement (of either type). Due to the unavailability of a closed-form solution, joint optimisation of inspection and maintenance policies resulting in the minimum total expected cost is performed using exhaustive search and genetic algorithm (GA), both with integer inspection period constraint, and a GA implementation with quasi-continuous inspection period. Although both exhaustive search and GA with integer inspection period provide identical results, the genetic algorithm presents a more efficient procedure and requires less computational time, which becomes more noticeable with increasing complexity of the problem, as in the case of GA with quasi-continuous inspection period. Based on the simulation results, some insights are made regarding the system's operation and cost optimisation. Expressions are derived for the expected number of system failures in terms of the cost ratio and component failure intensity. In addition, a criterion is derived for establishing acceptable level of expected system failures over the system's lifecycle. This can be useful when designing the system or analysing its performance.