## 2 Imperfect Maintenance Model  
The effect of maintenance activities on the state of a piece of equipment assumes three possibilities. One of them is the perfect maintenance, which leaves the equipment \"As Good as New\" (GAN). On the other side, we find the minimal maintenance that leaves the equipment in \"As Bad as Old\" (BAO) state. The third possibility is the imperfect maintenance which improves the state of the equipment by some degree, by an effectiveness rate or factor, and it is the one that better represents real-world situations. Several models have been developed to simulate imperfect maintenance. The Proportional Age Set-back (PAS) model proposed by [PERSON] et al. [12] and the Proportional Age Reduction (PAR) proposed by [PERSON] [13] are some of them. In this paper, we use an age reduction preventive maintenance model based on PAR. In this approach, each maintenance activity is assumed to shift the origin of time from which the age of the component had its last maintenance intervention. The PAR model assumes that the maintenance activity reduces proportionally by an effectiveness factor \(\varepsilon\) the age gained from previous maintenance, where \(\varepsilon\) ranges in the interval \(\varepsilon\in[0,1]\). This model contains the extreme situations BAO and GAN. The PAR model is reduced to a BAO situation when \(\varepsilon=0\), while \(\varepsilon=1\) reduces to a GAN situation. We will assume that a component is continuously monitored, so that no preventive action is taken until it arrives at a critical age. Let us denote this critical age by \(w_{c}\). The component is left as BAO when a corrective maintenance (CM) activity is executed. The execution of a preventive maintenance activity (PM) means the application of an effectiveness \(\varepsilon\) and a PAR aging model. The behavior of the system is shown in Figure 2.  
It can be observed that a \(\varepsilon\) proportion of the age gained since the last PM activity is shifted, and that time between two consecutive PM activities, \(M\), gets shorter. As \(M\) tends to zero, it is necessary to fix a limit value, \(M_{min}\), which indicates that the component should be upgraded. The model developed here considers an upgrading of \(\varepsilon=1\) (GAN). Figure 2: Age vs. chronological time in a Proportional Age Reduction (PAR) model under a Corrective Maintenance (CM) strategy. Under these conditions, it is possible to model the time between two consecutive maintenance activities, M. The time \(M_{1}\) between the installation of the component and the execution of the first activity is given by:  
\[M_{1}=t_{1}=w_{c}. \tag{1}\]  
We will denote by \(w_{1}^{+}\) the age of the component immediately after the first maintenance:  
\[w_{1}^{+}=(1-\varepsilon)t_{1}=(1-\varepsilon)w_{c}. \tag{2}\]  
Let \(t_{2}\) be the time in which the second maintenance activity is executed, and \(M_{2}\) the time interval between the first and the second maintenance activities:  
\[M_{2}=t_{2}-t_{1}=w_{c}\varepsilon. \tag{3}\]  
We will denote by \(w_{2}^{-}\) the age of the component immediately before the second maintenance activity:  
\[w_{2}^{-}=w_{1}^{+}+(t_{2}-t_{1}). \tag{4}\]  
In addition, taking into account expressions (1) and (4), the expression for \(t_{2}\) is obtained:  
\[t_{2}=w_{c}+w_{c}\cdot\varepsilon=w_{c}\cdot(1+\varepsilon). \tag{5}\]  
Proceeding in a similar way with \(t_{3}\) (the time in which the third maintenance activity is executed), and \(M_{3}\) (time interval between the second and the third maintenance activities), we have:  
\[ \begin{cases}w_{2}^{+}=(1-\varepsilon)t_{2}=(1-\varepsilon)(1+\varepsilon)w_ {c},\\ w_{3}^{-}=w_{2}^{+}+(t_{3}-t_{2}),\\ M_{3}=(t_{3}-t_{2})=w_{c}\cdot\varepsilon^{2}.\end{cases}  \tag{6}\]  
Thus, the generalization to \(t_{m}\), the time in which the \(m\)th maintenance activity is executed, and \(M_{m}\) the time interval between the \((m-1)\)th and the \(m\)th maintenance activities is given by:  
\[