Abstract
This paper develops models for the maintenance of a system based on np control charts with respect to the sampling interval. At any given time, the system is assumed to be in one of the three possible states; in-control, out-of-control and failure. If the control chart signals, suggesting the possibility of an out-of-control state, an investigation will be carried out. We assume that this investigation is perfect in that it reveals the true state of the system. If an assignable cause is confirmed by the investigation, a minor repair will be carried out to remove the cause. If the assignable cause is not attended to, it will gradually develop into a failure. When a failure occurs, the system cannot operate and a major repair is needed. We discuss three models depending on the assumptions related to the renewal mechanism, the occurrence of failures, and the time between minor repairs. The paper seeks to optimise the performance of such a system in terms of the sampling interval. Geometric processes are utilised for modelling the lifetimes between minor repairs if the minor repair cannot bring the system back to an as good as new condition. The expected cost per unit time for maintaining the systems with respect to the sampling interval of the control chart is obtained. Numerical examples are conducted to demonstrate the applicability of the methodology derived.
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Acknowledgements
This research is partially supported by the Engineering and Physical Sciences Research Council (EPSRC, UK) under grant numbers EP/C54658X/1 and EP/F038526/1.
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Appendix
Appendix
Since now all minor repair cycles are identical, then we have,
where P is the probability for a cycle ended up with a minor repair.
Substitute A1 and A2 into Equation (13) we have
where E 3′(T)=E 3(T)/(1−P), E 2′(T)=E 2(T)/P and E 2(T) and E 3(T) are given by Equations (8) and (9).
As 
and 
we finally have,
The same also applies to the cost formulation and therefore the resulting formula for E(h) is the same as Equation (7).
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Wang, W. Maintenance models based on the np control charts with respect to the sampling interval. J Oper Res Soc 62, 124–133 (2011). https://doi.org/10.1057/jors.2009.165
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DOI: https://doi.org/10.1057/jors.2009.165