## 3 Cost and Benefit Models
### Cost Related to the Production Speed Loss Because of Aging  
Traditionally, the literature assumes that the production rate or speed of the equipment is predetermined and constant during the equipment's life. However, it is expected that the production speed decreases as a consequence of the aging of the equipment. In [10], both linear and bent deterioration or aging speeds are proposed. As bent relationships between chronological time and age or deterioration level have not been proposed in the literature for modeling equipment speed loss, a quadratic relationship between the age and the production speed is proposed in this research. The production speed after having performed the \(m\)th maintenance activity can be calculated as:  
\[V_{m}(w)=V_{0}-\tau^{2}\cdot w_{m}(t,\varepsilon), \tag{24}\]where \(V_{0}\) is the initial production speed (i.e., as per design), \(\tau\) represents the speed reduction coefficient, and \(w_{m}(t,\varepsilon)\) is the age of the component after maintenance number \(m\), which under a PAR model is given by Equation (9).  
An average value of the production speed between the activities \((m-1)\) (\(V_{m-1}^{+}\)) and \(m\) (\(V_{m}^{-}\)) is required in order to determine the cost of the lost production speed. Let \(V_{m}^{*}\) be this average value and it can be calculated using the following equations:  
\[\left\{\begin{array}{rcl}V_{m}&=&V_{0}-\tau^{2}\cdot w_{c},\\ V_{m-1}^{+}&=&V_{0}-\tau^{2}\cdot w_{c}(1-\varepsilon)\cdot\sum_{k=1}^{n-1} \varepsilon^{k-1},\\ V_{m}^{*}&=&\frac{1}{M_{m}}\int_{w_{m-1}^{-}}^{w_{m}^{+}}v(t) \cdot dt=V_{0}-\tau^{2}\cdot w_{c}\left(\frac{1+(1-\varepsilon)\cdot\sum_{k=1 }^{n-1}\varepsilon^{k-1}}{2}\right).\end{array}\right. \tag{25}\]  
Similar to the failure rate, it is possible again to get an averaged production speed for the lifetime of a component, \(V^{*}\):  
\[V^{*}=\frac{\sum_{1}^{\varepsilon}V_{e}^{*}M_{e}}{M_{e}}. \tag{26}\]  
In addition, using expression (26) to calculate the average value of the speed production, it is possible to determine the production time lost \(t_{sl}\) related to a reduced speed. Assuming that only a fraction of the production system is available, the production time lost is expressed as:  
\[t_{sl}(\mathbf{x})=\left(1-A_{s}(\mathbf{x})\cdot\frac{V^{*}}{V_{0}}\right) \cdot L, \tag{27}\]  
\(A_{s}(\mathbf{x})\) being the availability of the system obtained as:  
\[A_{s}(\mathbf{x})=1-U_{s}(\mathbf{x}). \tag{28}\]  
\(U_{s}(\mathbf{x})\) is the unavailability of the system. It is evaluated using the system fault tree and the single component unavailability. These contributions are the time-dependent unreliability \(u_{r}(\mathbf{x})\) given by Equation (21) and the unavailability due to corrective maintenance \(u_{cm}(\mathbf{x})\) given by:  
\[u_{cm}(\mathbf{x})=\frac{1}{M}\cdot u_{r}(\mathbf{x})\cdot d_{cm}. \tag{29}\]  
The unavailability associated with preventive maintenance in the period \(L\) is given by:  
\[u_{pm}(\mathbf{x})=\frac{1}{M}\cdot d_{pm}. \tag{30}\]  
The cost related to production speed loss of the equipment (\(C_{sl}\)) in the period \(L\) is proportional to the production time lost and is given by:  
\[C_{sl}(\mathbf{x})=c_{hsl}\cdot t_{sl}(\mathbf{x}), \tag{31}\]  
\(c_{hsl}\) being the average hourly cost due to non-produced items and \(t_{sl}\) is defined in (27).