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Top1. Introduction
Studies maintenance systems just like the other areas of dependability (Avižienis, Laprie, & Randell, 2001) are becoming an increasingly essential in any type of production. Henceforth, the purpose of maintenance is no longer content with a simple cleaning or repair components. It is nowadays a variety of activities and much more complex in order to improve and increase the lifetime of the system.
With the permanent technological evolutions, systems have become more critical and sensitive to failures. These failures and malfunctions require increased maintenance to ensure an optimal level of service in terms of reliability, availability and cost. To remedy the failures of these systems, several maintenance policies have been proposed (Bris, Châtelet, & Yalaoui, 2003; Caldeira Duarte, Taborda, Craveiro, & Trigo, 2006; Remy, Corset, Despéraux, Doyen, & Gaudoin, 2013; Hagmark, 2011; Taghipour & Banjevic, 2011; Tsai, Wang, & Tsai, 2004; Vatn & Aven, 2010; Wang, 2002) according to their complexity and criticality.
Generally, strategies are grouped into two broad classes: the corrective (CM) and preventive maintenance (PM). Corrective maintenance is performed after a failure and is intended to put the system in working condition but that will not avoid the consequences of failure. A more defensive approach is to implement a preventive maintenance which is carried out when the system is operating and is intended to reduce and prevent these failures. The preventive maintenance can be performed at predetermined intervals (Planned maintenance) or according to prescribed criteria (Condition-based preventive maintenance) for assessing the system degradation state and decide on an intervention when a certain threshold is reached (Dijoux & Gaudoin, 2013; Remy et al., 2013).
After restarting the system, it is unknown if the next maintenance will be corrective or preventive. The simple way of modelling this situation is the competing risks theory, introduced in the context of maintenance in Bunea and Bedford (2002).
The idea of competing risks is to model the situation where a random event may be due to several causes or risks. The event can be failure mode, maintenance actions, etc. The competing risk theory has spread through various fields of science such as statistics, medicine, demography and actuarial science.
In the context of maintenance, the competing risks can be viewed as the maintenance actions like in Bedford and Alkali (2009) and Hagmark (2011). These maintenances can put the system in the state as good as new called AGAN (perfect maintenance) or in the state as bad as old called ABAO (minimal maintenance). However, reality is between these two extreme cases: maintenance reduces failures intensity but does not leave the system as good as new. This is known as imperfect maintenance (Remy et al., 2013).
In this study, the authors propose to define maintenance policies through the warning delivered by the system when the degradation passes a critical threshold and avoid the failure by a preventive maintenance. The approach is defined by the new dependant competing risks model called: Alert Delay model. This model is important for making decision about which kind of maintenance could be perform. When both kinds of maintenance are considered, it is necessary to add information about the possible dependency between PM and CM. Indeed, the aim of PM is to reduce the frequency of failure, so PM should delay CM. Finally, CM can have an influence on the future PM policy. So, to model this situation, it is needed to define dependent competing risks models.
The Alert Delay model provides this information, considering that maintenances are assumed to be perfect. However, despite this limit, this model offers a powerful framework to estimate the dependency between the two maintenances. In this case, we assumed that when the system fails, the system is immediately repaired and put in function again; so, each maintenance leaves the system in a state as good as new.