ABSTRACT
However, these models of dependence can only represent a few range of possible dependence. Moreover, when the system contains a large number of components, it seems that the dependence are hardly to be builded in the system level. In other hands, copulas are widely used to model the dependence structure of random variables as they separate dependence structure from marginal properties. Hong, Zhou, Zhang, & Ye (2014) model the dependence structure of stochastically deteriorating components by the ordinary copula which has several drawbacks when applied to Lévy processes and pointed out by Cont & Tankov (2004). To overcome these drawbacks, they extended the Sklar’s Theorem for Lévy measure and they defined a new copula for Lévy processes so called Lévy copulas. Kallsen & Tankov (2006) enrich the properties of Lévy copulas. Based-on marginal Gamma processes, we use Clayton-Lévy copula to model the dependent stochastic degradation of components. This model of dependence is different from previous literatures as Lévy copula provides a means of modeling stochastic dependence which can befrom independence to total dependence and it is easily simulated. Economic dependence offers opportunities to group maintenance actions so as to save set-up cost. Including the economic dependence,
1 INTRODUCTION
In order to ensure the productivity, the safety and to save the cost, maintenance policies have been widely studied since the last century. Nevertheless, the nowadays systems contain more and more components resulting in more complex structures. As a result, the research of maintenance policies focusing on the single-unit systems is not suitable any more. For a multi-component system, interactions between components have to be taken into consideration. According to Thomas (1986), generally interactions between components in a multicomponent system are classified into three types: economic dependence, stochastic dependence and structural dependence. In this paper, we consider economic and stochastic dependence. Stochastic dependence arises in industry due to the common environment or to the fact that components share the same stress. Hong, Zhou, Zhang, & Ye (2014) shows that maintenance optimization will be affected by dependent degradation. Golmakani & Moakedi (2012) propose a periodic inspection optimization model for a two-component repairable system with failure interactions, Mercier & Pham (2012) model dependence by a bivariate wear subordinator with Gamma marginal distributions.
